Embibe Experts Solutions for Chapter: Sequences and Series, Exercise 1: Exercise-1

If $a, b, c$ are in A.P., then show that:

$b+c-a, c+a-b, a+b-c$ are in A.P.

Sum the following series:

$1+\frac{3}{4}+\frac{7}{16}+\frac{15}{64}+\frac{31}{256}+\dots ..$ to infinity.

Using the relation A.M.$\ge$G.M., prove that $\left(a+b\right)·\left(b+c\right)·\left(c+a\right)>abc;$ if $a, b, c$ are positive real numbers.

Find the sum to $n$ terms of the sequence $3+33+333+3333+\dots \dots$ up to $n$ terms.

Find the sum to $n$-terms of the sequence

$1·3·2^2+2·4·3^2+3·5·4^2+\dots$

If $t_n=n\left(n+2\right)$, then find $\sum _{n=1}^k{t}_n$

If $S$ be the sum to infinity of a G.P., whose first term is $a$, then the sum of first $n$ terms is

The sum of the series $S=1^2-2^2+3^2-4^2+\dots -{2002}^2+{2003}^2$ is