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Câu 15 trang 109 SGK Đại số và Giải tích 11 Nâng cao

Cho dãy số (un) xác định bởi
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Cho dãy số (un) xác định bởi

\({u_1} = 3\,\text{ và }\,{u_{n + 1}} = {u_n} + 5\) với mọi \(n ≥ 1\).

LG a

Hãy tính u2, u4 và u6.

Lời giải chi tiết:

Ta có:

\(\eqalign{
& {u_2} = {u_1} + 5 = 8 \cr 
& {u_3} = {u_2} + 5 = 13 \cr 
& {u_4} = {u_3} + 5 = 18 \cr 
& {u_5} = {u_4} + 5 = 23 \cr 
& {u_6} = {u_5} + 5 = 28 \cr} \)

LG b

Chứng minh rằng \(u_n= 5n – 2\) với mọi \(n ≥ 1\).

Lời giải chi tiết:

Ta sẽ chứng minh : \(u_n= 5n – 2\) (1) với mọi \(n \in \mathbb N^*\), bằng phương pháp qui nạp. +) Với \(n = 1\), ta có \(u_1= 3 = 5.1 – 2\) Vậy (1) đúng khi \(n = 1\). +) Giả sử (1) đúng với \(n = k, k\in \mathbb N^*\), tức là: \(u_k=5k-2\) +) Ta sẽ chứng minh (1) cũng đúng khi \(n = k + 1\)

Thật vậy, từ công thức xác định dãy số (un) và giả thiết qui nạp ta có :

\({u_{k + 1}} = {u_k} + 5 \) \(= 5k - 2 + 5 = 5\left( {k + 1} \right) - 2\) Do đó (1) đúng với mọi \(n \in \mathbb N^*\).

Cách khác:

Ta có:

\(\begin{array}{l}
{u_n} = {u_{n - 1}} + 5\\
{u_{n - 1}} = {u_{n - 2}} + 5\\
...\\
{u_3} = {u_2} + 5\\
{u_2} = {u_1} + 5\\
\Rightarrow {u_n} + {u_{n - 1}} + ... + {u_3} + {u_2}\\
= \left( {{u_{n - 1}} + 5} \right) + \left( {{u_{n - 2}} + 5} \right) + ...\\
+ \left( {{u_2} + 5} \right) + \left( {{u_1} + 5} \right)\\
\Rightarrow {u_n} + {u_{n - 1}} + ... + {u_3} + {u_2}\\
= {u_{n - 1}} + {u_{n - 2}} + ... + {u_2} + {u_1}\\
+ \left( {5 + 5 + ... + 5 + 5} \right)(\text{ n-1 số 5})\\
\Rightarrow {u_n} = {u_1} + 5.\left( {n - 1} \right)\\
\Rightarrow {u_n} = 3 + 5n - 5 = 5n - 2
\end{array}\)

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