Chapter 9

If the sum of first \(n\) natural numbers is \(\frac{1}{5}\) times the sum of their squares, then find the value of \(n\).

If \(a_{1}=1\) and \(a_{n+1}=\frac{4+3 a_{n}}{3+2 a_{n}}, n \geq 1\) and if \(\lim _{n \rightarrow \infty} a_{n}=a\), then find the value of \(a\).

Prove that each number is the square of an odd integer in the sequence \(49,4489,444889, \ldots \ldots \ldots \ldots .\) in which every number is formed by inserting 48 in the middle of the previous number as indicated.

Let \(a_{1}, a_{2}, a_{3}, \cdots \cdots, a_{11}\) be real numbers satisfying \(a_{1}=15,27-2 a_{2}>0\) and \(a_{k}=2 a_{k-1}-a_{k-2}\) for \(k=3,4, \cdots \cdots, 11 .\) If \(\frac{a_{1}^{2}+a_{2}^{2}+\cdots \cdots+a_{11}^{2}}{11}=90\), then find the value of \(\frac{a_{1}+a_{2}+\cdots \cdots+a_{11}}{11}\).

If \(a>b>0\) and \(n \in N\), prove that \(a^{n}-b^{n} \geq n(a-b)(a b)^{\frac{n-1}{2}}\).

Given \(|x|<1\), sum to infinite terms \(\frac{1}{(1-x)\left(1-x^{3}\right)}+\frac{x^{2}}{\left(1-x^{3}\right)\left(1-x^{5}\right)}+\frac{x^{4}}{\left(1-x^{5}\right)\left(1-x^{7}\right)}+\cdots \cdots\)

Let \(f_{1}(x)=\frac{x}{3}+10\) for all \(x \in R\) and \(f_{n}(x)=f_{1}\left(f_{n-1}(x)\right)\) for \(n \geq 2 .\) Then find \(f_{n}(x)\).