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Câu 47 trang 123 SGK Đại số và Giải tích 11 Nâng cao

Trong các dãy số dưới đây
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Trong các dãy số dưới đây, dãy số nào là cấp số cộng, dãy số nào là cấp số nhân ? Hãy xác định công sai hoặc công bội của mỗi cấp số đó.

LG a

Dãy số (un) với un = 8n + 3

Phương pháp giải:

Xét hiệu \(u_{n+1}-u_n\) hoặc thương \(\frac{{{u_{n + 1}}}}{{{u_n}}}\). Nếu hiệu trên là hằng số thì dãy là CSC. Nếu thương trên là hằng số thì dãy là CSN.

Lời giải chi tiết:

Ta có: \({u_{n + 1}} - {u_n}\) \(= 8\left( {n + 1} \right) + 3 - \left( {8n + 3} \right) \) \( = 8n + 8 + 3 - 8n - 3\) \(= 8,\forall n \ge 1\)

Suy ra (un) là cấp số cộng với công sai \(d = 8\)

LG b

Dãy số (un) với \({u_n} = {n^2} + n + 1\)

Lời giải chi tiết:

Ta có: \({u_{n + 1}} - {u_n} \) \(= {\left( {n + 1} \right)^2} + \left( {n + 1} \right) + 1 - ({n^2} + n + 1) \) \( = {n^2} + 2n + 1 + n + 1 + 1 - {n^2} - n - 1 \) \(= 2n + 2\) \(= 2\left( {n + 1} \right)\) không là hằng số

Vậy (u­n) không là cấp số cộng.

\({{{u_{n + 1}}} \over {{u_n}}} = \frac{{{{\left( {n + 1} \right)}^2} + \left( {n + 1} \right) + 1}}{{{n^2} + n + 1}} \) \(= \frac{{{n^2} + 2n + 1 + n + 1 + 1}}{{{n^2} + n + 1}}\)

\( = {{{n^2} + 3n + 3} \over {{n^2} + n + 1}}\) không là hằng số nên (un) không là cấp số nhân.

Cách giải thích khác:

Ta có:

\(\begin{array}{l}
{u_1} = {1^2} + 1 + 1 = 3\\
{u_2} = {2^2} + 2 + 1 = 7\\
{u_3} = {3^2} + 3 + 1 = 13\\
\Rightarrow {u_2} - {u_1} = 4 \ne 6 = {u_3} - {u_2}
\end{array}\)

Do đó dãy không là CSC. Lại có: \(\frac{{{u_2}}}{{{u_1}}} = \frac{7}{3} \ne \frac{{13}}{7} = \frac{{{u_3}}}{{{u_2}}}\) Do đó dãy không là CSN.

LG c

Dãy số (un) với \({u_n} = {3.8^n}\)

Lời giải chi tiết:

\({{{u_{n + 1}}} \over {{u_n}}} = {{{{3.8}^{n + 1}}} \over {{{3.8}^n}}} = 8,\forall n \ge 1.\)

Do đó (un) là cấp số nhân với công bội \(q = 8\).

LG d

Dãy số (un) với \({u_n} = \left( {n + 2} \right){.3^n}\)

Lời giải chi tiết:

\({u_{n + 1}} - {u_n}\) \(= \left( {n + 3} \right){.3^{n + 1}} - \left( {n + 2} \right){3^n} \)

\(= {3^n}\left( {3n + 9 - n - 2} \right) = \left( {2n + 7} \right){3^n}\) không là hằng số nên (un) không là cấp số cộng.

\({{{u_{n + 1}}} \over {{u_n}}} = {{\left( {n + 3} \right){{.3}^{n + 1}}} \over {\left( {n + 2} \right){{.3}^n}}} = {{3n + 9} \over {n + 2}}\) không là hằng số nên (un) không là cấp số nhân.

Cách khác:

\(\begin{array}{l}
{u_1} = \left( {1 + 2} \right){.3^1} = 9\\
{u_2} = \left( {2 + 2} \right){.3^2} = 36\\
{u_3} = \left( {3 + 2} \right){.3^3} = 135\\
\Rightarrow {u_2} - {u_1} = 27 \ne 99 = {u_3} - {u_2}
\end{array}\)

Do đó dãy không là CSC. Lại có: \(\frac{{{u_2}}}{{{u_1}}} = \frac{{36}}{9} = 4 \ne \frac{{135}}{{36}} = \frac{{{u_3}}}{{{u_2}}}\) Do đó dãy không là CSN.

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