JEE Mains · Maths · STD 11 - 8. sequence and series
Let \(\left\{a_{n}\right\}_{n=0}^{\infty}\) be a sequence such that \(a _{0}= a _{1}=0\) and \(a _{ n +2}=2 a _{ n +1}- a _{ n }+1\) for all \(n \geq 0\). Then, \(\sum\limits_{ n =2}^{\infty} \frac{ a _{ n }}{7^{ n }}\) is equal to
- A \(\frac{6}{343}\)
- B \(\frac{7}{216}\)
- C \(\frac{8}{343}\)
- D \(\frac{49}{216}\)
Answer & Solution
Correct Answer
(B) \(\frac{7}{216}\)
Step-by-step Solution
Detailed explanation
\(a_{2}=1, a_{3}=3 a_{4}=6\) \(a_{n}=\frac{n(n-1)}{2}\) \(S=\sum\limits_{n=2}^{\infty} \frac{n(n-1)}{2\left(7^{n}\right)}\) \(S=\frac{1}{7^{2}}+\frac{3}{7^{3}}+\frac{6}{7^{4}}+\frac{10}{7^{5}}+\frac{15}{7^{5}}+\ldots \ldots\)…
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