{"id":609006,"date":"2026-09-14T09:12:37","date_gmt":"2026-09-14T02:12:37","guid":{"rendered":"https:\/\/hoanghamobile.com\/tin-tuc\/?p=609006"},"modified":"2026-09-14T09:12:37","modified_gmt":"2026-09-14T02:12:37","slug":"cong-thuc-cap-so-nhan","status":"publish","type":"post","link":"https:\/\/hoanghamobile.com\/tin-tuc\/cong-thuc-cap-so-nhan\/","title":{"rendered":"C\u00f4ng th\u1ee9c c\u1ea5p s\u1ed1 nh\u00e2n: s\u1ed1 h\u1ea1ng t\u1ed5ng qu\u00e1t, t\u1ed5ng v\u00e0 b\u00e0i t\u1eadp c\u00f3 l\u1eddi gi\u1ea3i"},"content":{"rendered":"<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\"><strong>C\u00f4ng th\u1ee9c c\u1ea5p s\u1ed1 nh\u00e2n<\/strong> l\u00e0 d\u00e3y s\u1ed1 m\u00e0 k\u1ec3 t\u1eeb s\u1ed1 h\u1ea1ng th\u1ee9 hai, m\u1ed7i s\u1ed1 h\u1ea1ng b\u1eb1ng s\u1ed1 h\u1ea1ng \u0111\u1ee9ng ngay tr\u01b0\u1edbc n\u00f3 nh\u00e2n v\u1edbi m\u1ed9t s\u1ed1 kh\u00f4ng \u0111\u1ed5i q g\u1ecdi l\u00e0 c\u00f4ng b\u1ed9i. <\/span><b>C\u00f4ng th\u1ee9c c\u1ea5p s\u1ed1 nh\u00e2n<\/b><span style=\"font-weight: 400;\"> quan tr\u1ecdng nh\u1ea5t l\u00e0 s\u1ed1 h\u1ea1ng t\u1ed5ng qu\u00e1t: u(n) = u(1) \u00d7 q^(n-1). Nh\u1edd \u0111\u00f3 b\u1ea1n t\u00ecm \u0111\u01b0\u1ee3c s\u1ed1 h\u1ea1ng b\u1ea5t k\u1ef3 khi bi\u1ebft u(1) v\u00e0 q. D\u01b0\u1edbi \u0111\u00e2y l\u00e0 tr\u1ecdn b\u1ed9 c\u00f4ng th\u1ee9c l\u1edbp 11: \u0111\u1ecbnh ngh\u0129a, s\u1ed1 h\u1ea1ng t\u1ed5ng qu\u00e1t, t\u00ednh ch\u1ea5t, t\u1ed5ng n s\u1ed1 h\u1ea1ng \u0111\u1ea7u, t\u1ed5ng l\u00f9i v\u00f4 h\u1ea1n, so s\u00e1nh v\u1edbi c\u1ea5p s\u1ed1 c\u1ed9ng, 6 b\u00e0i t\u1eadp c\u00f3 l\u1eddi gi\u1ea3i v\u00e0 c\u00e1c l\u1ed7i hay m\u1eafc.<\/span><\/p>\n<h2 style=\"text-align: justify;\"><b>C\u1ea5p s\u1ed1 nh\u00e2n l\u00e0 g\u00ec?<\/b><\/h2>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">C\u1ea5p s\u1ed1 nh\u00e2n l\u00e0 d\u00e3y s\u1ed1 (u(n)) th\u1ecfa m\u00e3n u(n) = u(n-1) \u00d7 q v\u1edbi m\u1ecdi n l\u1edbn h\u01a1n ho\u1eb7c b\u1eb1ng 2, trong \u0111\u00f3 h\u1eb1ng s\u1ed1 q g\u1ecdi l\u00e0 c\u00f4ng b\u1ed9i. T\u1ec9 s\u1ed1 hai s\u1ed1 h\u1ea1ng li\u00ean ti\u1ebfp lu\u00f4n kh\u00f4ng \u0111\u1ed5i: q = u(n) \/ u(n-1). D\u1ef1a v\u00e0o c\u00f4ng b\u1ed9i, c\u1ea5p s\u1ed1 nh\u00e2n c\u00f3 th\u1ec3 t\u0103ng, gi\u1ea3m ho\u1eb7c thay \u0111\u1ed5i d\u1ea5u gi\u1eefa c\u00e1c s\u1ed1 h\u1ea1ng. \u0110\u00e2y l\u00e0 m\u1ed9t d\u1ea1ng d\u00e3y s\u1ed1 quan tr\u1ecdng, th\u01b0\u1eddng xu\u1ea5t hi\u1ec7n trong c\u00e1c b\u00e0i to\u00e1n v\u1ec1 s\u1ed1 h\u1ea1ng, t\u1ed5ng v\u00e0 s\u1ef1 t\u0103ng tr\u01b0\u1edfng theo c\u1ea5p s\u1ed1.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">V\u00ed d\u1ee5 d\u00e3y 3, 6, 12, 24, 48 c\u00f3 t\u1ec9 s\u1ed1 lu\u00f4n b\u1eb1ng 2, l\u00e0 c\u1ea5p s\u1ed1 nh\u00e2n v\u1edbi u(1) = 3 v\u00e0 q = 2. D\u00e3y 3, -6, 12, -24 c\u00f3 q = -2 n\u00ean \u0111\u1ed5i d\u1ea5u xen k\u1ebd. N\u1ebfu q = 1, c\u00e1c s\u1ed1 h\u1ea1ng s\u1ebd kh\u00f4ng thay \u0111\u1ed5i; c\u00f2n khi q = 0, t\u1eeb s\u1ed1 h\u1ea1ng th\u1ee9 2 tr\u1edf \u0111i \u0111\u1ec1u b\u1eb1ng 0.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">\u0110i\u1ec1u ki\u1ec7n \u0111\u1ec3 m\u1ed9t d\u00e3y l\u00e0 c\u1ea5p s\u1ed1 nh\u00e2n:<\/span><\/p>\n<ul style=\"text-align: justify;\">\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">u(1) kh\u00e1c 0: n\u1ebfu u(1) = 0 th\u00ec m\u1ecdi s\u1ed1 h\u1ea1ng sau \u0111\u1ec1u b\u1eb1ng 0.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">q kh\u00e1c 0: n\u1ebfu q = 0 th\u00ec t\u1ec9 s\u1ed1 u(n) \/ u(n-1) kh\u00f4ng x\u00e1c \u0111\u1ecbnh.<\/span><\/li>\n<\/ul>\n<p><img fetchpriority=\"high\" decoding=\"async\" class=\"aligncenter size-full wp-image-609426\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/09\/cong-thuc-cap-so-nhan-1.jpg\" alt=\"cong-thuc-cap-so-nhan-1\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/09\/cong-thuc-cap-so-nhan-1.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/09\/cong-thuc-cap-so-nhan-1-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/09\/cong-thuc-cap-so-nhan-1-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<h2 style=\"text-align: justify;\"><b>C\u00f4ng th\u1ee9c c\u1ea5p s\u1ed1 nh\u00e2n h\u1ea1ng t\u1ed5ng qu\u00e1t<\/b><\/h2>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">V\u1edbi s\u1ed1 h\u1ea1ng \u0111\u1ea7u u(1) v\u00e0 c\u00f4ng b\u1ed9i q:<\/span><\/p>\n<p style=\"text-align: justify;\"><b>u(n) = u(1) \u00d7 q^(n-1)<\/b><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">C\u00e1ch hi\u1ec3u: T\u1eeb u(1) \u0111\u1ebfn u(n) ta nh\u00e2n v\u1edbi q \u0111\u00fang n &#8211; 1 l\u1ea7n, n\u00ean s\u1ed1 m\u0169 l\u00e0 n &#8211; 1 ch\u1ee9 kh\u00f4ng ph\u1ea3i n.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">V\u00ed d\u1ee5: u(1) = 5, q = 3 th\u00ec u(6) = 5 \u00d7 3^5 = 5 \u00d7 243 = 1215.<\/span><\/p>\n<table>\n<tbody>\n<tr>\n<td><b>K\u00fd hi\u1ec7u<\/b><\/td>\n<td><b>\u00dd ngh\u0129a<\/b><\/td>\n<td><b>Ghi ch\u00fa<\/b><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">u(1)<\/span><\/td>\n<td><span style=\"font-weight: 400;\">S\u1ed1 h\u1ea1ng \u0111\u1ea7u<\/span><\/td>\n<td><span style=\"font-weight: 400;\">Kh\u00e1c 0<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">q<\/span><\/td>\n<td><span style=\"font-weight: 400;\">C\u00f4ng b\u1ed9i<\/span><\/td>\n<td><span style=\"font-weight: 400;\">Kh\u00e1c 0, c\u00f3 th\u1ec3 \u00e2m<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">n<\/span><\/td>\n<td><span style=\"font-weight: 400;\">V\u1ecb tr\u00ed s\u1ed1 h\u1ea1ng c\u1ea7n t\u00ecm<\/span><\/td>\n<td><span style=\"font-weight: 400;\">Nguy\u00ean d\u01b0\u01a1ng<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">u(n)<\/span><\/td>\n<td><span style=\"font-weight: 400;\">S\u1ed1 h\u1ea1ng th\u1ee9 n<\/span><\/td>\n<td><span style=\"font-weight: 400;\">K\u1ebft qu\u1ea3 c\u1ea7n t\u00ednh<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 style=\"text-align: justify;\"><b>T\u00ednh ch\u1ea5t c\u1ee7a c\u00f4ng th\u1ee9c c\u1ea5p s\u1ed1 nh\u00e2n<\/b><\/h2>\n<p style=\"text-align: justify;\"><b>u(k)\u00b2 = u(k-1) \u00d7 u(k+1), v\u1edbi m\u1ecdi k l\u1edbn h\u01a1n ho\u1eb7c b\u1eb1ng 2.<\/b><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Ch\u1ee9ng minh: v\u00ec u(k) = u(k-1) \u00d7 q v\u00e0 u(k+1) = u(k) \u00d7 q n\u00ean u(k-1) \u00d7 u(k+1) = u(k-1) \u00d7 u(k) \u00d7 q = u(k) \u00d7 u(k) = u(k)\u00b2. V\u00ed d\u1ee5 v\u1edbi 2, 6, 18: c\u00f3 6\u00b2 = 36 v\u00e0 2 \u00d7 18 = 36.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">\u0110\u00e2y l\u00e0 c\u00f4ng c\u1ee5 \u0111\u1ec3 ch\u1ee9ng minh ba s\u1ed1 l\u1eadp th\u00e0nh c\u1ea5p s\u1ed1 nh\u00e2n: v\u1edbi ba s\u1ed1 a, b, c kh\u00e1c 0, ch\u1ec9 c\u1ea7n ki\u1ec3m tra b\u00b2 = a \u00d7 c. Ng\u01b0\u1ee3c l\u1ea1i, khi \u0111\u1ec1 cho bi\u1ebft ba s\u1ed1 l\u1eadp th\u00e0nh c\u1ea5p s\u1ed1 nh\u00e2n, h\u00e3y vi\u1ebft ngay b\u00b2 = a \u00d7 c \u0111\u1ec3 t\u00ecm \u1ea9n r\u1ed3i th\u1eed l\u1ea1i nghi\u1ec7m.<\/span><\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-609427\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/09\/cong-thuc-cap-so-nhan-2.jpg\" alt=\"cong-thuc-cap-so-nhan-2\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/09\/cong-thuc-cap-so-nhan-2.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/09\/cong-thuc-cap-so-nhan-2-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/09\/cong-thuc-cap-so-nhan-2-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<h2 style=\"text-align: justify;\"><b>C\u00f4ng th\u1ee9c t\u00ednh t\u1ed5ng n s\u1ed1 h\u1ea1ng \u0111\u1ea7u<\/b><\/h2>\n<p style=\"text-align: justify;\"><b>G\u1ecdi S(n) = u(1) + u(2) + &#8230; + u(n).<\/b><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Khi q kh\u00e1c 1: S(n) = u(1) \u00d7 (1 &#8211; q^n) \/ (1 &#8211; q)<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">D\u1ea1ng t\u01b0\u01a1ng \u0111\u01b0\u01a1ng (nh\u00e2n t\u1eed v\u00e0 m\u1eabu v\u1edbi -1):<\/span><\/p>\n<p style=\"text-align: justify;\"><b>S(n) = u(1) \u00d7 (q^n &#8211; 1) \/ (q &#8211; 1)<\/b><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">V\u00ed d\u1ee5: u(1) = 2, q = 3 th\u00ec S(6) = 2 \u00d7 (3^6 &#8211; 1) \/ (3 &#8211; 1) = 728. C\u1ed9ng tr\u1ef1c ti\u1ebfp 2 + 6 + 18 + 54 + 162 + 486 c\u0169ng \u0111\u01b0\u1ee3c 728.<\/span><\/p>\n<p style=\"text-align: justify;\"><b>Tr\u01b0\u1eddng h\u1ee3p ri\u00eang q = 1: S(n) = n \u00d7 u(1)<\/b><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Ph\u1ea3i t\u00e1ch ri\u00eang v\u00ec khi q = 1 th\u00ec m\u1eabu th\u1ee9c 1 &#8211; q b\u1eb1ng 0. L\u00fac \u0111\u00f3 m\u1ecdi s\u1ed1 h\u1ea1ng \u0111\u1ec1u b\u1eb1ng u(1) n\u00ean t\u1ed5ng l\u00e0 n l\u1ea7n u(1). V\u00ed d\u1ee5 u(1) = 7, q = 1 th\u00ec S(5) = 5 \u00d7 7 = 35.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">V\u1eady khi \u0111\u1ec1 ch\u01b0a cho bi\u1ebft q, h\u00e3y x\u00e9t c\u1ea3 hai kh\u1ea3 n\u0103ng, gi\u1ed1ng th\u00f3i quen bi\u1ec7n lu\u1eadn \u1edf b\u00e0i <\/span><a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/cach-tinh-can-bac-2\/\"><span style=\"font-weight: 400;\">c\u00e1ch t\u00ednh c\u0103n b\u1eadc 2<\/span><\/a><span style=\"font-weight: 400;\">.\u00a0<\/span><\/p>\n<h2 style=\"text-align: justify;\"><b>T\u1ed5ng c\u1ee7a c\u1ea5p s\u1ed1 nh\u00e2n l\u00f9i v\u00f4 h\u1ea1n<\/b><\/h2>\n<p style=\"text-align: justify;\"><b>C\u00f4ng th\u1ee9c c\u1ea5p s\u1ed1 nh\u00e2n<\/b><span style=\"font-weight: 400;\"> l\u00f9i v\u00f4 h\u1ea1n l\u00e0 c\u1ea5p s\u1ed1 nh\u00e2n v\u00f4 h\u1ea1n c\u00f3 |q| &lt; 1, t\u1ee9c -1 &lt; q &lt; 1 v\u00e0 q kh\u00e1c 0. T\u1ed5ng c\u1ee7a n\u00f3 l\u00e0:<\/span><\/p>\n<p style=\"text-align: justify;\"><b>S = u(1) \/ (1 &#8211; q)<\/b><\/p>\n<p style=\"text-align: justify;\"><b>\u0110i\u1ec1u ki\u1ec7n |q| &lt; 1 l\u00e0 b\u1eaft bu\u1ed9c<\/b><span style=\"font-weight: 400;\">. Ch\u1ec9 khi \u0111\u00f3 q^n m\u1edbi ti\u1ebfn v\u1ec1 0 v\u00e0 t\u1ed5ng m\u1edbi h\u1ed9i t\u1ee5 v\u1ec1 gi\u00e1 tr\u1ecb h\u1eefu h\u1ea1n. N\u1ebfu |q| l\u1edbn h\u01a1n ho\u1eb7c b\u1eb1ng 1, c\u00f4ng th\u1ee9c kh\u00f4ng \u00e1p d\u1ee5ng \u0111\u01b0\u1ee3c.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">V\u00ed d\u1ee5: d\u00e3y 1, 1\/2, 1\/4, 1\/8, &#8230; c\u00f3 q = 1\/2 n\u00ean S = 1 \/ (1 &#8211; 1\/2) = 2. Ng\u01b0\u1ee3c l\u1ea1i, d\u00e3y 1, 2, 4, 8, &#8230; c\u00f3 q = 2 n\u00ean t\u1ed5ng v\u00f4 h\u1ea1n kh\u00f4ng t\u1ed3n t\u1ea1i.<\/span><\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-609428\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/09\/cong-thuc-cap-so-nhan-3.jpg\" alt=\"cong-thuc-cap-so-nhan-3\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/09\/cong-thuc-cap-so-nhan-3.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/09\/cong-thuc-cap-so-nhan-3-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/09\/cong-thuc-cap-so-nhan-3-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<h2 style=\"text-align: justify;\"><b>Ph\u00e2n bi\u1ec7t c\u00f4ng th\u1ee9c c\u1ea5p s\u1ed1 nh\u00e2n v\u00e0 c\u1ea5p s\u1ed1 c\u1ed9ng<\/b><\/h2>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">C\u1ea5p s\u1ed1 c\u1ed9ng v\u00e0 c\u1ea5p s\u1ed1 nh\u00e2n \u0111\u1ec1u m\u00f4 t\u1ea3 m\u1ed9t d\u00e3y s\u1ed1 c\u00f3 quy lu\u1eadt, nh\u01b0ng c\u00e1ch c\u00e1c s\u1ed1 h\u1ea1ng thay \u0111\u1ed5i l\u1ea1i ho\u00e0n to\u00e0n kh\u00e1c nhau. S\u1ef1 kh\u00e1c bi\u1ec7t n\u00e0y th\u01b0\u1eddng g\u00e2y nh\u1ea7m l\u1eabn khi nh\u1eadn d\u1ea1ng ho\u1eb7c \u00e1p d\u1ee5ng c\u00f4ng th\u1ee9c v\u00e0o b\u00e0i t\u1eadp. Ph\u1ea7n d\u01b0\u1edbi \u0111\u00e2y s\u1ebd \u0111\u1eb7t 2 d\u1ea1ng d\u00e3y s\u1ed1 c\u1ea1nh nhau \u0111\u1ec3 l\u00e0m r\u00f5 \u0111\u1eb7c \u0111i\u1ec3m c\u1ea7n ch\u00fa \u00fd.<\/span><\/p>\n<table>\n<tbody>\n<tr>\n<td><b>Ti\u00eau ch\u00ed<\/b><\/td>\n<td><b>C\u1ea5p s\u1ed1 c\u1ed9ng<\/b><\/td>\n<td><b>C\u1ea5p s\u1ed1 nh\u00e2n<\/b><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">Quy lu\u1eadt<\/span><\/td>\n<td><span style=\"font-weight: 400;\">u(n) = u(n-1) + d<\/span><\/td>\n<td><span style=\"font-weight: 400;\">u(n) = u(n-1) \u00d7 q<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">H\u1eb1ng s\u1ed1<\/span><\/td>\n<td><span style=\"font-weight: 400;\">C\u00f4ng sai d<\/span><\/td>\n<td><span style=\"font-weight: 400;\">C\u00f4ng b\u1ed9i q<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">Nh\u1eadn bi\u1ebft<\/span><\/td>\n<td><span style=\"font-weight: 400;\">Hi\u1ec7u kh\u00f4ng \u0111\u1ed5i<\/span><\/td>\n<td><span style=\"font-weight: 400;\">T\u1ec9 s\u1ed1 kh\u00f4ng \u0111\u1ed5i<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">S\u1ed1 h\u1ea1ng t\u1ed5ng qu\u00e1t<\/span><\/td>\n<td><span style=\"font-weight: 400;\">u(n) = u(1) + (n-1) \u00d7 d<\/span><\/td>\n<td><span style=\"font-weight: 400;\">u(n) = u(1) \u00d7 q^(n-1)<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">Ba s\u1ed1 li\u00ean ti\u1ebfp<\/span><\/td>\n<td><span style=\"font-weight: 400;\">2 \u00d7 u(k) = u(k-1) + u(k+1)<\/span><\/td>\n<td><span style=\"font-weight: 400;\">u(k)\u00b2 = u(k-1) \u00d7 u(k+1)<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">T\u1ed5ng n s\u1ed1 h\u1ea1ng \u0111\u1ea7u<\/span><\/td>\n<td><span style=\"font-weight: 400;\">S(n) = n \u00d7 (u(1) + u(n)) \/ 2<\/span><\/td>\n<td><span style=\"font-weight: 400;\">S(n) = u(1) \u00d7 (1 &#8211; q^n) \/ (1 &#8211; q)<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">T\u1ed5ng v\u00f4 h\u1ea1n<\/span><\/td>\n<td><span style=\"font-weight: 400;\">Kh\u00f4ng t\u1ed3n t\u1ea1i<\/span><\/td>\n<td><span style=\"font-weight: 400;\">T\u1ed3n t\u1ea1i khi |q| &lt; 1<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">G\u1eb7p d\u00e3y l\u1ea1, h\u00e3y l\u1ea5y hi\u1ec7u hai s\u1ed1 h\u1ea1ng li\u00ean ti\u1ebfp: hi\u1ec7u kh\u00f4ng \u0111\u1ed5i l\u00e0 c\u1ea5p s\u1ed1 c\u1ed9ng. Hi\u1ec7u thay \u0111\u1ed5i th\u00ec l\u1ea5y t\u1ec9 s\u1ed1, t\u1ec9 s\u1ed1 kh\u00f4ng \u0111\u1ed5i l\u00e0 c\u1ea5p s\u1ed1 nh\u00e2n. Xem th\u00eam b\u00e0i<\/span><a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/cong-thuc-tinh-tong-day-so-cach-deu\/\"> <span style=\"font-weight: 400;\">c\u00f4ng th\u1ee9c t\u00ednh t\u1ed5ng d\u00e3y s\u1ed1 c\u00e1ch \u0111\u1ec1u<\/span><\/a><span style=\"font-weight: 400;\">.\u00a0<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-609429\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/09\/cong-thuc-cap-so-nhan-4.jpg\" alt=\"cong-thuc-cap-so-nhan-4\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/09\/cong-thuc-cap-so-nhan-4.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/09\/cong-thuc-cap-so-nhan-4-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/09\/cong-thuc-cap-so-nhan-4-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<h2 style=\"text-align: justify;\"><b>B\u00e0i t\u1eadp v\u1eadn d\u1ee5ng c\u00f3 l\u1eddi gi\u1ea3i c\u00f4ng th\u1ee9c c\u1ea5p s\u1ed1 nh\u00e2n<\/b><\/h2>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Sau khi n\u1eafm \u0111\u01b0\u1ee3c \u0111\u1eb7c \u0111i\u1ec3m v\u00e0 c\u00f4ng th\u1ee9c c\u1ee7a t\u1eebng d\u1ea1ng, vi\u1ec7c l\u00e0m b\u00e0i t\u1eadp s\u1ebd gi\u00fap c\u1ee7ng c\u1ed1 ki\u1ebfn th\u1ee9c hi\u1ec7u qu\u1ea3 h\u01a1n. C\u00e1c b\u00e0i t\u1eadp d\u01b0\u1edbi \u0111\u00e2y \u0111\u01b0\u1ee3c s\u1eafp x\u1ebfp theo m\u1ee9c \u0111\u1ed9 ph\u00f9 h\u1ee3p, k\u1ebft h\u1ee3p gi\u1eefa nh\u1eadn bi\u1ebft quy lu\u1eadt v\u00e0 \u00e1p d\u1ee5ng c\u00f4ng th\u1ee9c. Ph\u1ea7n l\u1eddi gi\u1ea3i \u0111i k\u00e8m gi\u00fap ng\u01b0\u1eddi h\u1ecdc theo d\u00f5i c\u00e1ch th\u1ef1c hi\u1ec7n v\u00e0 t\u1ef1 ki\u1ec3m tra k\u1ebft qu\u1ea3.<\/span><\/p>\n<p style=\"text-align: justify;\"><b>B\u00e0i 1: T\u00ecm s\u1ed1 h\u1ea1ng khi c\u00f4ng b\u1ed9i \u00e2m<\/b><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">\u0110\u1ec1: Cho u(1) = 3 v\u00e0 q = -2. T\u00ednh u(7).<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Gi\u1ea3i: u(7) = 3 \u00d7 (-2)^6. S\u1ed1 m\u0169 6 ch\u1eb5n n\u00ean (-2)^6 = 64, v\u1eady u(7) = 192.<\/span><\/p>\n<p style=\"text-align: justify;\"><b>B\u00e0i 2: T\u00ecm c\u00f4ng b\u1ed9i v\u00e0 s\u1ed1 h\u1ea1ng \u0111\u1ea7u<\/b><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">\u0110\u1ec1: Cho u(3) = 12 v\u00e0 u(5) = 48. T\u00ecm u(1) v\u00e0 q.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Gi\u1ea3i: u(1) \u00d7 q\u00b2 = 12 v\u00e0 u(1) \u00d7 q^4 = 48. L\u1eadp t\u1ec9 s\u1ed1 v\u1ebf theo v\u1ebf \u0111\u1ec3 kh\u1eed u(1): q\u00b2 = 4, suy ra q = 2 ho\u1eb7c q = -2. C\u1ea3 hai \u0111\u1ec1u cho u(1) = 12 \/ 4 = 3, n\u00ean b\u00e0i c\u00f3 hai \u0111\u00e1p s\u1ed1.<\/span><\/p>\n<p style=\"text-align: justify;\"><b>B\u00e0i 3: T\u00ednh t\u1ed5ng n s\u1ed1 h\u1ea1ng \u0111\u1ea7u<\/b><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">\u0110\u1ec1: Cho u(1) = 2 v\u00e0 q = 2. T\u00ednh S(8).<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Gi\u1ea3i: V\u00ec q kh\u00e1c 1 n\u00ean S(8) = 2 \u00d7 (2^8 &#8211; 1) \/ (2 &#8211; 1) = 2 \u00d7 255 = 510.<\/span><\/p>\n<p style=\"text-align: justify;\"><b>B\u00e0i 4: T\u1ed5ng c\u1ea5p s\u1ed1 nh\u00e2n l\u00f9i v\u00f4 h\u1ea1n<\/b><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">\u0110\u1ec1: T\u00ednh t\u1ed5ng c\u1ee7a d\u00e3y 6, 2, 2\/3, 2\/9, &#8230;<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Gi\u1ea3i: q = 2 \/ 6 = 1\/3, ki\u1ec3m tra c\u1eb7p sau (2\/3) \/ 2 = 1\/3 n\u00ean t\u1ec9 s\u1ed1 kh\u00f4ng \u0111\u1ed5i. V\u00ec |1\/3| &lt; 1 n\u00ean S = 6 \/ (1 &#8211; 1\/3) = 9.<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-609430\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/09\/cong-thuc-cap-so-nhan-5.jpg\" alt=\"cong-thuc-cap-so-nhan-5\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/09\/cong-thuc-cap-so-nhan-5.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/09\/cong-thuc-cap-so-nhan-5-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/09\/cong-thuc-cap-so-nhan-5-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<p style=\"text-align: justify;\"><b>B\u00e0i 5: Ch\u1ee9ng minh ba s\u1ed1 l\u1eadp th\u00e0nh c\u1ea5p s\u1ed1 nh\u00e2n<\/b><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">\u0110\u1ec1: T\u00ecm x \u0111\u1ec3 ba s\u1ed1 x, x + 3, x + 9 theo th\u1ee9 t\u1ef1 l\u1eadp th\u00e0nh c\u1ea5p s\u1ed1 nh\u00e2n.<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Gi\u1ea3i: \u0110i\u1ec1u ki\u1ec7n l\u00e0 (x + 3)\u00b2 = x \u00d7 (x + 9), t\u1ee9c x\u00b2 + 6x + 9 = x\u00b2 + 9x. R\u00fat g\u1ecdn: 6x + 9 = 9x, suy ra x = 3. Th\u1eed l\u1ea1i: ba s\u1ed1 l\u00e0 3, 6, 12 v\u00e0 6\u00b2 = 3 \u00d7 12 = 36. V\u1eady x = 3.<\/span><\/p>\n<p style=\"text-align: justify;\"><b>B\u00e0i 6: B\u00e0i to\u00e1n l\u00e3i k\u00e9p<\/b><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">\u0110\u1ec1: M\u1ed9t ng\u01b0\u1eddi g\u1eedi 100 tri\u1ec7u \u0111\u1ed3ng theo l\u00e3i k\u00e9p, l\u00e3i su\u1ea5t 6% m\u1ed9t n\u0103m kh\u00f4ng \u0111\u1ed5i. Sau 5 n\u0103m ng\u01b0\u1eddi \u0111\u00f3 c\u00f3 bao nhi\u00eau ti\u1ec1n?<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Gi\u1ea3i: G\u1ecdi A(n) l\u00e0 s\u1ed1 ti\u1ec1n sau n n\u0103m. M\u1ed7i n\u0103m s\u1ed1 ti\u1ec1n \u0111\u01b0\u1ee3c nh\u00e2n v\u1edbi 1,06 n\u00ean \u0111\u00e2y l\u00e0 c\u1ea5p s\u1ed1 nh\u00e2n c\u00f3 q = 1,06 v\u00e0 A(n) = 100 \u00d7 1,06^n (tri\u1ec7u \u0111\u1ed3ng). V\u1edbi n = 5: 1,06^5 \u2248 1,338226 n\u00ean A(5) \u2248 133,82 tri\u1ec7u \u0111\u1ed3ng.<\/span><\/p>\n<h2 style=\"text-align: justify;\"><b>Nh\u1eefng l\u1ed7i th\u01b0\u1eddng g\u1eb7p khi l\u00e0m b\u00e0i c\u1ea5p s\u1ed1 nh\u00e2n<\/b><\/h2>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Khi l\u00e0m b\u00e0i <\/span><b>c\u00f4ng th\u1ee9c c\u1ea5p s\u1ed1 nh\u00e2n<\/b><span style=\"font-weight: 400;\">, nhi\u1ec1u l\u1ed7i kh\u00f4ng xu\u1ea5t ph\u00e1t t\u1eeb vi\u1ec7c kh\u00f4ng nh\u1edb c\u00f4ng th\u1ee9c m\u00e0 do b\u1ecf s\u00f3t \u0111i\u1ec1u ki\u1ec7n ho\u1eb7c \u00e1p d\u1ee5ng c\u00f4ng th\u1ee9c ch\u01b0a \u0111\u00fang tr\u01b0\u1eddng h\u1ee3p. \u0110\u1eb7c bi\u1ec7t, c\u00e1c b\u00e0i t\u00ednh t\u1ed5ng, t\u00ecm c\u00f4ng b\u1ed9i v\u00e0 x\u00e1c \u0111\u1ecbnh s\u1ed1 h\u1ea1ng th\u01b0\u1eddng c\u00f3 nh\u1eefng chi ti\u1ebft nh\u1ecf d\u1ec5 g\u00e2y m\u1ea5t \u0111i\u1ec3m. Ki\u1ec3m tra l\u1ea1i \u0111i\u1ec1u ki\u1ec7n tr\u01b0\u1edbc khi thay s\u1ed1 s\u1ebd gi\u00fap h\u1ea1n ch\u1ebf nh\u1eefng sai s\u00f3t n\u00e0y.<\/span><\/p>\n<ul style=\"text-align: justify;\">\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Qu\u00ean x\u00e9t q = 1 khi t\u00ednh t\u1ed5ng: C\u00f4ng th\u1ee9c S(n) = u(1) \u00d7 (1 &#8211; q^n) \/ (1 &#8211; q) c\u00f3 m\u1eabu b\u1eb1ng 0 n\u1ebfu q = 1. Khi \u0111\u00f3 ph\u1ea3i d\u00f9ng S(n) = n \u00d7 u(1).<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">B\u1ecf qua \u0111i\u1ec1u ki\u1ec7n |q| &lt; 1: C\u00f4ng th\u1ee9c S = u(1) \/ (1 &#8211; q) ch\u1ec9 d\u00f9ng cho c\u1ea5p s\u1ed1 nh\u00e2n l\u00f9i v\u00f4 h\u1ea1n. N\u1ebfu |q| &gt;= 1, t\u1ed5ng v\u00f4 h\u1ea1n kh\u00f4ng h\u1ed9i t\u1ee5 theo c\u00f4ng th\u1ee9c n\u00e0y.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Nh\u1ea7m s\u1ed1 m\u0169 n v\u1edbi n &#8211; 1: S\u1ed1 h\u1ea1ng th\u1ee9 n l\u00e0 u(1) \u00d7 q^(n-1). \u0110\u00e2y l\u00e0 l\u1ed7i th\u01b0\u1eddng g\u1eb7p khi thay gi\u00e1 tr\u1ecb n v\u00e0o c\u00f4ng th\u1ee9c s\u1ed1 h\u1ea1ng t\u1ed5ng qu\u00e1t.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Nh\u1ea7m v\u1edbi c\u1ea5p s\u1ed1 c\u1ed9ng: Th\u1ea5y d\u00e3y t\u0103ng \u0111\u1ec1u l\u00e0 v\u1ed9i d\u00f9ng c\u00f4ng th\u1ee9c c\u1ed9ng. C\u1ea7n ki\u1ec3m tra t\u1ec9 s\u1ed1 gi\u1eefa hai s\u1ed1 h\u1ea1ng li\u00ean ti\u1ebfp tr\u01b0\u1edbc khi x\u00e1c \u0111\u1ecbnh \u0111\u00f3 l\u00e0 c\u1ea5p s\u1ed1 nh\u00e2n hay c\u1ea5p s\u1ed1 c\u1ed9ng.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Qu\u00ean nghi\u1ec7m \u00e2m c\u1ee7a c\u00f4ng b\u1ed9i v\u00e0 sai d\u1ea5u khi q \u00e2m: T\u1eeb q\u00b2 = 4 ph\u1ea3i nh\u1eadn c\u1ea3 q = 2 v\u00e0 q = -2. V\u1edbi q \u00e2m, q^(n-1) d\u01b0\u01a1ng khi n &#8211; 1 ch\u1eb5n v\u00e0 \u00e2m khi n &#8211; 1 l\u1ebb, v\u00ec v\u1eady c\u1ea7n ki\u1ec3m tra d\u1ea5u c\u1ee7a t\u1eebng s\u1ed1 h\u1ea1ng.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">B\u1ecf qu\u00ean \u0111i\u1ec1u ki\u1ec7n u(1) kh\u00e1c 0 v\u00e0 q kh\u00e1c 0: Khi gi\u1ea3i b\u00e0i b\u1eb1ng t\u1ec9 s\u1ed1 q = u(n) \/ u(n-1), c\u1ea7n ki\u1ec3m tra \u0111i\u1ec1u ki\u1ec7n \u0111\u1ec3 ph\u00e9p chia c\u00f3 ngh\u0129a. Sau khi t\u00ecm \u0111\u01b0\u1ee3c nghi\u1ec7m, n\u00ean thay ng\u01b0\u1ee3c v\u00e0o d\u1eef ki\u1ec7n ban \u0111\u1ea7u \u0111\u1ec3 ki\u1ec3m tra l\u1ea1i k\u1ebft qu\u1ea3. H\u00e3y gi\u1eef th\u00f3i quen th\u1eed l\u1ea1i nghi\u1ec7m nh\u01b0 \u1edf b\u00e0i<\/span><a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/cong-thuc-chinh-hop\/\"> <span style=\"font-weight: 400;\">c\u00f4ng th\u1ee9c ch\u1ec9nh h\u1ee3p<\/span><\/a><span style=\"font-weight: 400;\">.<\/span><\/li>\n<\/ul>\n<h2 style=\"text-align: justify;\"><b>K\u1ebft lu\u1eadn<\/b><\/h2>\n<p style=\"text-align: justify;\"><b>C\u00f4ng th\u1ee9c c\u1ea5p s\u1ed1 nh\u00e2n<\/b><span style=\"font-weight: 400;\"> xoay quanh b\u1ed1n c\u00f4ng th\u1ee9c: u(n) = u(1) \u00d7 q^(n-1); u(k)\u00b2 = u(k-1) \u00d7 u(k+1); S(n) = u(1) \u00d7 (1 &#8211; q^n) \/ (1 &#8211; q) v\u1edbi tr\u01b0\u1eddng h\u1ee3p ri\u00eang S(n) = n \u00d7 u(1) khi q = 1; v\u00e0 S = u(1) \/ (1 &#8211; q) khi |q| &lt; 1. Nh\u1edb \u0111i\u1ec1u ki\u1ec7n \u00e1p d\u1ee5ng quan tr\u1ecdng ngang v\u1edbi nh\u1edb c\u00f4ng th\u1ee9c. H\u00e3y luy\u1ec7n l\u1ea1i 6 b\u00e0i t\u1eadp tr\u00ean v\u00e0 ki\u1ec3m tra k\u1ebft qu\u1ea3 b\u1eb1ng c\u00e1ch li\u1ec7t k\u00ea v\u00e0i s\u1ed1 h\u1ea1ng \u0111\u1ea7u.<\/span><\/p>\n<h2 style=\"text-align: justify;\"><b>C\u00e2u h\u1ecfi th\u01b0\u1eddng g\u1eb7p v\u1ec1 c\u1ea5p s\u1ed1 nh\u00e2n<\/b><\/h2>\n<h3 style=\"text-align: justify;\"><b>C\u00f4ng th\u1ee9c c\u1ea5p s\u1ed1 nh\u00e2n l\u00e0 g\u00ec?<\/b><\/h3>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">C\u1ea5p s\u1ed1 nh\u00e2n l\u00e0 d\u00e3y s\u1ed1 m\u00e0 k\u1ec3 t\u1eeb s\u1ed1 h\u1ea1ng th\u1ee9 hai, m\u1ed7i s\u1ed1 h\u1ea1ng b\u1eb1ng s\u1ed1 h\u1ea1ng \u0111\u1ee9ng ngay tr\u01b0\u1edbc n\u00f3 nh\u00e2n v\u1edbi m\u1ed9t s\u1ed1 kh\u00f4ng \u0111\u1ed5i q g\u1ecdi l\u00e0 c\u00f4ng b\u1ed9i, v\u1edbi u(1) kh\u00e1c 0 v\u00e0 q kh\u00e1c 0. V\u00ed d\u1ee5 d\u00e3y 3, 6, 12, 24, 48 c\u00f3 q = 2.<\/span><\/p>\n<h3 style=\"text-align: justify;\"><b>C\u00f4ng th\u1ee9c s\u1ed1 h\u1ea1ng t\u1ed5ng qu\u00e1t c\u1ee7a c\u1ea5p s\u1ed1 nh\u00e2n l\u00e0 g\u00ec?<\/b><\/h3>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">S\u1ed1 h\u1ea1ng t\u1ed5ng qu\u00e1t l\u00e0 u(n) = u(1) \u00d7 q^(n-1). S\u1ed1 m\u0169 l\u00e0 n &#8211; 1 v\u00ec t\u1eeb u(1) \u0111\u1ebfn u(n) ta nh\u00e2n v\u1edbi q \u0111\u00fang n &#8211; 1 l\u1ea7n.<\/span><\/p>\n<h3 style=\"text-align: justify;\"><b>C\u00f4ng th\u1ee9c t\u00ednh t\u1ed5ng n s\u1ed1 h\u1ea1ng \u0111\u1ea7u l\u00e0 g\u00ec?<\/b><\/h3>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Khi q kh\u00e1c 1, S(n) = u(1) \u00d7 (1 &#8211; q^n) \/ (1 &#8211; q), ho\u1eb7c d\u1ea1ng t\u01b0\u01a1ng \u0111\u01b0\u01a1ng S(n) = u(1) \u00d7 (q^n &#8211; 1) \/ (q &#8211; 1).<\/span><\/p>\n<h3 style=\"text-align: justify;\"><b>N\u1ebfu c\u00f4ng b\u1ed9i q = 1 th\u00ec t\u00ednh t\u1ed5ng th\u1ebf n\u00e0o?<\/b><\/h3>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">Khi q = 1, m\u1eabu th\u1ee9c 1 &#8211; q b\u1eb1ng 0 n\u00ean ph\u1ea3i d\u00f9ng c\u00f4ng th\u1ee9c ri\u00eang S(n) = n \u00d7 u(1), v\u00ec l\u00fac \u0111\u00f3 m\u1ecdi s\u1ed1 h\u1ea1ng \u0111\u1ec1u b\u1eb1ng u(1).<\/span><\/p>\n<h3 style=\"text-align: justify;\"><b>C\u00f4ng th\u1ee9c c\u1ea5p s\u1ed1 nh\u00e2n l\u00f9i v\u00f4 h\u1ea1n l\u00e0 g\u00ec?<\/b><\/h3>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">\u0110\u00f3 l\u00e0 c\u1ea5p s\u1ed1 nh\u00e2n v\u00f4 h\u1ea1n c\u00f3 |q| nh\u1ecf h\u01a1n 1, t\u1ed5ng b\u1eb1ng S = u(1) \/ (1 &#8211; q). \u0110i\u1ec1u ki\u1ec7n |q| nh\u1ecf h\u01a1n 1 l\u00e0 b\u1eaft bu\u1ed9c v\u00ec ch\u1ec9 khi \u0111\u00f3 t\u1ed5ng m\u1edbi h\u1ed9i t\u1ee5.<\/span><\/p>\n<h3 style=\"text-align: justify;\"><b>L\u00e0m sao ch\u1ee9ng minh ba s\u1ed1 l\u1eadp th\u00e0nh c\u1ea5p s\u1ed1 nh\u00e2n?<\/b><\/h3>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">V\u1edbi ba s\u1ed1 a, b, c kh\u00e1c 0, ta ki\u1ec3m tra \u0111\u1eb3ng th\u1ee9c b\u00b2 = a \u00d7 c. V\u00ed d\u1ee5 6\u00b2 = 36 v\u00e0 2 \u00d7 18 = 36 n\u00ean 2, 6, 18 l\u1eadp th\u00e0nh c\u1ea5p s\u1ed1 nh\u00e2n.<\/span><\/p>\n<h3 style=\"text-align: justify;\"><b>C\u1ea5p s\u1ed1 nh\u00e2n kh\u00e1c c\u1ea5p s\u1ed1 c\u1ed9ng \u1edf \u0111i\u1ec3m n\u00e0o?<\/b><\/h3>\n<p style=\"text-align: justify;\"><span style=\"font-weight: 400;\">C\u1ea5p s\u1ed1 c\u1ed9ng c\u1ed9ng th\u00eam c\u00f4ng sai d v\u1edbi u(n) = u(1) + (n-1) \u00d7 d, c\u00f2n c\u1ea5p s\u1ed1 nh\u00e2n nh\u00e2n v\u1edbi c\u00f4ng b\u1ed9i q v\u00e0 u(n) = u(1) \u00d7 q^(n-1).<\/span><\/p>\n<p style=\"text-align: justify;\"><b>Xem th\u00eam:<\/b><\/p>\n<ul>\n<li>\n<p class=\"elementor-heading-title elementor-size-default\"><a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/giai-thua-la-gi\/\">Giai th\u1eeba l\u00e0 g\u00ec? C\u00f4ng th\u1ee9c n!, b\u1ea3ng gi\u00e1 tr\u1ecb v\u00e0 \u1ee9ng d\u1ee5ng<\/a><\/p>\n<\/li>\n<li>\n<p class=\"elementor-heading-title elementor-size-default\"><a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/trong-luong-rieng-la-gi\/\">Tr\u1ecdng l\u01b0\u1ee3ng ri\u00eang l\u00e0 g\u00ec? C\u00f4ng th\u1ee9c d = P\/V v\u00e0 b\u1ea3ng tra c\u1ee9u<\/a><\/p>\n<\/li>\n<\/ul>\n<p><script type=\"application\/ld+json\">\n{\n \"@context\": \"https:\/\/schema.org\",\n \"@type\": \"FAQPage\",\n \"mainEntity\": [\n  {\n   \"@type\": \"Question\",\n   \"name\": \"C\u00f4ng th\u1ee9c c\u1ea5p s\u1ed1 nh\u00e2n l\u00e0 g\u00ec?\",\n   \"acceptedAnswer\": {\n    \"@type\": \"Answer\",\n    \"text\": \"C\u1ea5p s\u1ed1 nh\u00e2n l\u00e0 d\u00e3y s\u1ed1 m\u00e0 k\u1ec3 t\u1eeb s\u1ed1 h\u1ea1ng th\u1ee9 hai, m\u1ed7i s\u1ed1 h\u1ea1ng b\u1eb1ng s\u1ed1 h\u1ea1ng \u0111\u1ee9ng ngay tr\u01b0\u1edbc n\u00f3 nh\u00e2n v\u1edbi m\u1ed9t s\u1ed1 kh\u00f4ng \u0111\u1ed5i q g\u1ecdi l\u00e0 c\u00f4ng b\u1ed9i, v\u1edbi u(1) kh\u00e1c 0 v\u00e0 q kh\u00e1c 0. V\u00ed d\u1ee5 d\u00e3y 3, 6, 12, 24, 48 c\u00f3 q = 2.\"\n   }\n  },\n  {\n   \"@type\": \"Question\",\n   \"name\": \"C\u00f4ng th\u1ee9c s\u1ed1 h\u1ea1ng t\u1ed5ng qu\u00e1t c\u1ee7a c\u1ea5p s\u1ed1 nh\u00e2n l\u00e0 g\u00ec?\",\n   \"acceptedAnswer\": {\n    \"@type\": \"Answer\",\n    \"text\": \"S\u1ed1 h\u1ea1ng t\u1ed5ng qu\u00e1t l\u00e0 u(n) = u(1) \u00d7 q^(n-1). S\u1ed1 m\u0169 l\u00e0 n - 1 v\u00ec t\u1eeb u(1) \u0111\u1ebfn u(n) ta nh\u00e2n v\u1edbi q \u0111\u00fang n - 1 l\u1ea7n.\"\n   }\n  },\n  {\n   \"@type\": \"Question\",\n   \"name\": \"C\u00f4ng th\u1ee9c t\u00ednh t\u1ed5ng n s\u1ed1 h\u1ea1ng \u0111\u1ea7u l\u00e0 g\u00ec?\",\n   \"acceptedAnswer\": {\n    \"@type\": \"Answer\",\n    \"text\": \"Khi q kh\u00e1c 1, S(n) = u(1) \u00d7 (1 - q^n) \/ (1 - q), ho\u1eb7c d\u1ea1ng t\u01b0\u01a1ng \u0111\u01b0\u01a1ng S(n) = u(1) \u00d7 (q^n - 1) \/ (q - 1).\"\n   }\n  },\n  {\n   \"@type\": \"Question\",\n   \"name\": \"N\u1ebfu c\u00f4ng b\u1ed9i q = 1 th\u00ec t\u00ednh t\u1ed5ng th\u1ebf n\u00e0o?\",\n   \"acceptedAnswer\": {\n    \"@type\": \"Answer\",\n    \"text\": \"Khi q = 1, m\u1eabu th\u1ee9c 1 - q b\u1eb1ng 0 n\u00ean ph\u1ea3i d\u00f9ng c\u00f4ng th\u1ee9c ri\u00eang S(n) = n \u00d7 u(1), v\u00ec l\u00fac \u0111\u00f3 m\u1ecdi s\u1ed1 h\u1ea1ng \u0111\u1ec1u b\u1eb1ng u(1).\"\n   }\n  },\n  {\n   \"@type\": \"Question\",\n   \"name\": \"C\u00f4ng th\u1ee9c c\u1ea5p s\u1ed1 nh\u00e2n l\u00f9i v\u00f4 h\u1ea1n l\u00e0 g\u00ec?\",\n   \"acceptedAnswer\": {\n    \"@type\": \"Answer\",\n    \"text\": \"\u0110\u00f3 l\u00e0 c\u1ea5p s\u1ed1 nh\u00e2n v\u00f4 h\u1ea1n c\u00f3 |q| nh\u1ecf h\u01a1n 1, t\u1ed5ng b\u1eb1ng S = u(1) \/ (1 - q). \u0110i\u1ec1u ki\u1ec7n |q| nh\u1ecf h\u01a1n 1 l\u00e0 b\u1eaft bu\u1ed9c v\u00ec ch\u1ec9 khi \u0111\u00f3 t\u1ed5ng m\u1edbi h\u1ed9i t\u1ee5.\"\n   }\n  },\n  {\n   \"@type\": \"Question\",\n   \"name\": \"L\u00e0m sao ch\u1ee9ng minh ba s\u1ed1 l\u1eadp th\u00e0nh c\u1ea5p s\u1ed1 nh\u00e2n?\",\n   \"acceptedAnswer\": {\n    \"@type\": \"Answer\",\n    \"text\": \"V\u1edbi ba s\u1ed1 a, b, c kh\u00e1c 0, ta ki\u1ec3m tra \u0111\u1eb3ng th\u1ee9c b\u00b2 = a \u00d7 c. V\u00ed d\u1ee5 6\u00b2 = 36 v\u00e0 2 \u00d7 18 = 36 n\u00ean 2, 6, 18 l\u1eadp th\u00e0nh c\u1ea5p s\u1ed1 nh\u00e2n.\"\n   }\n  },\n  {\n   \"@type\": \"Question\",\n   \"name\": \"C\u1ea5p s\u1ed1 nh\u00e2n kh\u00e1c c\u1ea5p s\u1ed1 c\u1ed9ng \u1edf \u0111i\u1ec3m n\u00e0o?\",\n   \"acceptedAnswer\": {\n    \"@type\": \"Answer\",\n    \"text\": \"C\u1ea5p s\u1ed1 c\u1ed9ng c\u1ed9ng th\u00eam c\u00f4ng sai d v\u1edbi u(n) = u(1) + (n-1) \u00d7 d, c\u00f2n c\u1ea5p s\u1ed1 nh\u00e2n nh\u00e2n v\u1edbi c\u00f4ng b\u1ed9i q v\u00e0 u(n) = u(1) \u00d7 q^(n-1).\"\n   }\n  }\n ]\n}\n<\/script><\/p>\n","protected":false},"excerpt":{"rendered":"<p>C\u00f4ng th\u1ee9c c\u1ea5p s\u1ed1 nh\u00e2n l\u00e0 d\u00e3y s\u1ed1 m\u00e0 k\u1ec3 t\u1eeb s\u1ed1 h\u1ea1ng th\u1ee9 hai, m\u1ed7i s\u1ed1 h\u1ea1ng b\u1eb1ng s\u1ed1 h\u1ea1ng \u0111\u1ee9ng ngay tr\u01b0\u1edbc n\u00f3 nh\u00e2n v\u1edbi m\u1ed9t s\u1ed1 kh\u00f4ng \u0111\u1ed5i q g\u1ecdi l\u00e0 c\u00f4ng b\u1ed9i. C\u00f4ng th\u1ee9c c\u1ea5p s\u1ed1 nh\u00e2n quan tr\u1ecdng nh\u1ea5t l\u00e0 s\u1ed1 h\u1ea1ng t\u1ed5ng qu\u00e1t: u(n) = u(1) \u00d7 q^(n-1). Nh\u1edd [&hellip;]<\/p>\n","protected":false},"author":84,"featured_media":609425,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"c\u00f4ng th\u1ee9c c\u1ea5p s\u1ed1 nh\u00e2n","_yoast_wpseo_title":"C\u00f4ng Th\u1ee9c C\u1ea5p S\u1ed1 Nh\u00e2n: S\u1ed1 H\u1ea1ng, T\u1ed5ng V\u00e0 B\u00e0i T\u1eadp Gi\u1ea3i","_yoast_wpseo_metadesc":"C\u00f4ng th\u1ee9c c\u1ea5p s\u1ed1 nh\u00e2n g\u1ed3m s\u1ed1 h\u1ea1ng t\u1ed5ng qu\u00e1t, t\u1ed5ng n s\u1ed1 h\u1ea1ng \u0111\u1ea7u, t\u1ed5ng l\u00f9i v\u00f4 h\u1ea1n v\u00e0 6 b\u00e0i t\u1eadp c\u00f3 l\u1eddi gi\u1ea3i. Xem chi ti\u1ebft t\u1ea1i Ho\u00e0ng H\u00e0 Mobile.","footnotes":""},"categories":[29235,3910,30453],"tags":[],"class_list":["post-609006","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-goc-to-mo","category-kham-pha","category-kien-thuc"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v27.3 (Yoast SEO v27.4) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>C\u00f4ng Th\u1ee9c C\u1ea5p S\u1ed1 Nh\u00e2n: S\u1ed1 H\u1ea1ng, T\u1ed5ng V\u00e0 B\u00e0i T\u1eadp Gi\u1ea3i<\/title>\n<meta name=\"description\" content=\"C\u00f4ng th\u1ee9c c\u1ea5p s\u1ed1 nh\u00e2n g\u1ed3m s\u1ed1 h\u1ea1ng t\u1ed5ng qu\u00e1t, t\u1ed5ng n s\u1ed1 h\u1ea1ng \u0111\u1ea7u, t\u1ed5ng l\u00f9i v\u00f4 h\u1ea1n v\u00e0 6 b\u00e0i t\u1eadp c\u00f3 l\u1eddi gi\u1ea3i. 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