Chapter 9
Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry · 307 exercises
Problem 319
If the sum of first \(n\) natural numbers is \(\frac{1}{5}\) times the sum of their squares, then find the value of \(n\).
3 step solution
Problem 320
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 \(a\).
4 step solution
Problem 321
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.
3 step solution
Problem 323
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 \(\frac{a_{1}+a_{2}+\cdots \cdots+a_{11}}{11}\).
5 step solution
Problem 324
If \(a>b>0\) and \(n \in N\), prove that \(a^{n}-b^{n} \geq n(a-b)(a b)^{\frac{n-1}{2}}\).
4 step solution
Problem 325
Given \(|x|<1\), sum to infinite terms \(\frac{1}{(1-x)\left(1-x^{3}\right)}+\frac{x^{2}}{\left(1-x^{3}\right)\left(1-x^{5}\right)}+\frac{x^{4}}{\left(1-x^{5}\right)\left(1-x^{7}\right)}+\cdots \cdots\)
5 step solution
Problem 326
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 \(f_{n}(x)\).
4 step solution