KC Sinha Mathematics Solution Class 11 Chapter 19 Sum to n Terms of Special Series Exercise 19.2

 Exercise 19.2

Page no 19.20

Type 1

Question 1

Find the sum of the series $1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\ldots$ to $\infty$

Question 2

Prove that $1+\frac{1}{4}+\frac{1}{4^{2}}+\frac{1}{4^{3}}+\ldots$ to $\infty=\frac{4}{3}$

Question 3

Prove that $\left(2^{1 / 4}, 4^{1 / 8} \cdot 8^{1 / 16}, 16^{1 / 32} \ldots, 10 \infty\right)=2$

Question 4

The first term of an infinite G.t, is 2 and the sum to infinity is 6 . Find the common ratio.

Page no 19.21

Question 5

If $x=1+a+a^{2}+a^{3}+\ldots$ to $\operatorname{so}$, where comumon ratio $r+0<a<1$ show that $a=\frac{x-1}{x}$

Question 6

If each term of an infinite G.P. is twice the sum of all the succeeding terms, find the common ratio of G.R.

Question 7

Show that in an infinite G.P. with common ratio $r, 0<r<1$, any term is less than the sum of all the succeeding terms if $r>\frac{1}{2}$.

Question 8

If $b=a+a^{2}+a^{3}+\ldots$ to $\infty$ prove thin $a=\frac{b}{1+b}$.

Question 9

The sum of an infinite G.P. is 57 and the sum of their cubes is 9747 . find the common ratio of G.P.

Question 10

The sum of an infinite G.P. is 15 and the sum of square of the terms of G.P. is 45. find the first term of G.P.

Question 11

If $A=1+r^{a}+r^{2 a}+\ldots$ to $\infty$ and $B=1+r^{b}+r^{2 h}+\ldots$, to $\infty$, prove that

$r=\left(\frac{A-1}{A}\right)^{\mathrm{L} a}=\left(\frac{B-1}{B}\right)^{1 / 6}$

Question 12

If $-1<x<1$ and $-1<y<1$, find the sum to infinity of the series $x+y+\left(x^{2}+x y+y^{2}\right)+\left(x^{3}+x^{2} y+x y^{2}+y^{3}\right)+\ldots \infty$

Question 13

A square is given. A second square is made by joining the middle points of the sides of the first square. Then a third square is made by joining the middle points of the sides of the second square. This process is continued indefinitely. Show that the area of the first square is equal to the sum of areas of all the succeeding squares.

Question 14

Express $0.2 \overline{73}$ as a rational number.

Question 15

Express the following as rational number :

(i) $4,2 \overline{7}$

(ii) $0.2 \overline{53}$

Type 2

Question 16

Find the sum to infinity of the series $\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+\ldots$

Question 17

Find the sum to infinity of the series $1+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots$