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Câu 18 trang 109 SGK Đại số và Giải tích 11 Nâng cao

Cho dãy số (sn)
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Cho dãy số (sn🅺) với  \({s_n} = \sin \left( {4n - 1} \right){\pi \over 6}.\)

LG a

Chứng minh rằng \({s_n} = {s_{n + 3}}\) với mọi \(n ≥ 1\)

Lời giải chi tiết:

Với \(n>1\) tùy ý, ta có :

\(\eqalign{
& {s_{n + 3}} = \sin \left[ {4\left( {n + 3} \right) - 1} \right]{\pi \over 6} \cr 
& = \sin \left[ {4n - 1 + 12} \right]{\pi \over 6} \cr 
& = \sin \left[ {\left( {4n - 1} \right){\pi \over 6} + 2\pi } \right] \cr 
♏& = \sin \left( {4n - 1} \right){\pi \over 6} = {s_n} \cr} \)

LG b

Hãy tính tổng \(15\) số hạng đầu tiên của dãy số đã cho.

Lời giải chi tiết:

Từ kết quả phần a ta có :

\(\eqalign{
& {s_1} = {s_4} = {s_7} = {s_{10}} = {s_{13}}, \cr 
& {s_2} = {s_5} = {s_8} = {s_{11}} = {s_{14}}, \cr 
🐲& {s_3} = {s_6} = {s_9} = {s_{12}} = {s_{15}} \cr} \)

Từ đó suy ra : \({s_1} + {s_2} + {s_3} \) \(= {s_4} + {s_5}{ + s_6} \) \(= {s_7} + {s_8} + {s_9} \) \(= {s_{10}} + {s_{11}} + {s_{12}} \) \(= {s_{13}} + {s_{14}} + {s_{15}}\) Do đó: \({S_{15}} = {s_1} + {s_2} + ... + {s_{15}}\) \(=({s_1} + {s_2} + {s_3})\)+\(({s_4} + {s_5}{ + s_6})\)+...+\(( {s_{13}} + {s_{14}} + {s_{15}})\) \(= 5\left( {{s_1} + {s_2} + {s_3}} \right)\) Ta có:

\(\begin{array}{l}
{s_1} = \sin \left[ {\left( {4.1 - 1} \right).\frac{\pi }{6}} \right] = \sin \frac{\pi }{2} = 1\\
{s_2} = \sin \left[ {\left( {4.2 - 1} \right).\frac{\pi }{6}} \right] = \sin \frac{{7\pi }}{6}\\
= \sin \left( {\pi + \frac{\pi }{6}} \right) = - \sin \frac{\pi }{6} = - \frac{1}{2}\\
{s_3} = \sin \left[ {\left( {4.3 - 1} \right).\frac{\pi }{6}} \right] = \sin \frac{{11\pi }}{6}\\
= \sin \left( {2\pi - \frac{\pi }{6}} \right) = \sin \left( { - \frac{\pi }{6}} \right) = - \frac{1}{2}
\end{array}\)

Do đó \({s_1} = 1,{s_2} = - {1 \over 2}\,\text{ và }\,{s_3} = - {1 \over 2} \) \( \Rightarrow {s_1} + {s_2} + {s_3} = 1 - \frac{1}{2} - \frac{1}{2} = 0\) \(\Rightarrow {s_{15}} =5.0= 0\)

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