Embibe Experts Solutions for Chapter: Sequences and Series, Exercise 4: EXERCISE-4

If $a,b,c$are in H.P., $b,c,d$ are in G.P. $\&c,d,e$ are in A.P., then Show that $e=a{b}^2/(2a-b{)}^2$.

Prove that the sum of the infinite series $\frac{1·3}{2}+\frac{3·5}{2^2}+\frac{5·7}{2^3}+\frac{7·9}{2^4}+\dots \dots \dots .\infty =23$.

Find the sum to $n$ terms: $\frac{1}{x+1}+\frac{2x}{(x+1)(x+2)}+\frac{3x^2}{(x+1)(x+2)(x+3)}+\dots \dots .$

Find the sum to $n$ terms: $\frac{a_1}{1+a_1}+\frac{a_2}{\left(1+a_1\right)\left(1+a_2\right)}+\frac{a_3}{\left(1+a_1\right)\left(1+a_2\right)\left(1+a_3\right)}+\dots \dots .$

Prove that the number $\underset{n \mathrm{digits}}{\underbrace{4 4 4\dots \dots \dots ..4} }\underset{(n-1)\mathrm{digits} }{\underbrace{8 8 8\dots \dots \dots \dots .8} }9$ is a perfect square of the number
$\underset{(n-1)\mathrm{digits} }{\underbrace{6 6 6\dots \dots \dots \dots .6} 7}$

Find the $n^{\mathrm{th}}$ term and the sum to '$n$' terms of the series: $1+5+13+29+61+\dots \dots$

If $a,b,c$ are three positive real number then prove that :$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\ge \frac{3}{2}$

Prove that among all triangles with a given perimeter, the equilateral triangle has the maximum area.