KC Sinha Mathematics Solution Class 11 Chapter 19 Sum to n Terms of Special Series Exercise 19.1

 Exercise 19.1

Page no 19.15

Type 1

Question 

Find th1e sum of n t1erms of the series whose nth term is: 

1. $n(n-1)(n+1)$

2. $n\left(n^{2}+1\right)$

3. $n(n+3)$

4. $n^{3}-3^{n}$

Type 2

Question 

Find the sum of the following series to $n$ terms :

5. $1^{3}+3^{3}+5^{3}+\ldots$

6. $1^{2}+4^{2}+7^{2}+10^{2}+\ldots$

7. $1^{2}-2^{2}+3^{2}-4^{2}+\ldots$

8. $1.2 .3+2.3 .5+3.4 .7+\ldots$

9. $1+(1+3)+(1+3+5)+\ldots$

10. $1.2^{2}+2.3^{2}+3.4^{2}+\ldots$

II. $\left(n^{2}-1^{2}\right)+2\left(n^{2}-2^{2}\right)+3\left(n^{2}-3^{2}\right)+\ldots$

12. $3.1^{2}+5.2^{2}+7.3^{2}+\ldots$

13. $3.8+6.11+9.14+\ldots$

14. $1.2+2.3+3.4+4.5+\ldots$

Question 15

Find the $n$th term and hence the 20 th term of the series $2.4+4.6+6.8+\ldots$ Also find the sum of its 20 terms.

Question 16

Show that $\frac{1.2^{2}+2 \cdot 3^{2}+\ldots+n(n+1)^{2}}{1^{2} \cdot 2+2^{2} \cdot 3+\ldots+n^{2}(n+1)}=\frac{3 n+5}{3 n+1}$

Type 3

Question 17

Find $1+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots$ to $n$ terms.

$\left[\text { Hint }: t_{n}=\frac{1}{1+2+\ldots+n}=\frac{2}{n(n+1)}\right]$

Question 18

Find the sum to $n$ terms of the series

$\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+\ldots$

Question 19

Find $\frac{1}{1+1^{2}+1^{4}}+\frac{2}{1+2^{2}+2^{4}}+\frac{3}{1+3^{2}+3^{4}}+\ldots$ to $n$ terms

Question 20

Find the sum to infinity of the series

$\frac{1}{2.4}+\frac{1}{4.6}+\frac{1}{6.8}+\frac{1}{8.10}+\ldots$

Type 4

Question 21

Find the $n$th term and sum to $n$ terms of the following series :

(i) $2+6+12+20+\ldots$

(ii) $3+6+11+18+\ldots$

(iii) $3+15+35+63+\ldots$

(iv) $1+9+24+46+75+\ldots$

(v) $1+5+12+22+\ldots$

(vi) $3+7+13+21+31+\ldots$

Question 22

Find the sum to 10 terms of the series $1+3+6+10+\ldots$

Question 23

Natural numbers have been divided into groups in the following way : $1 ;(2,3) ;(4,5,6) ;(7,8,9,10)$; and so on.

Find the sum of numbers in the 50 th group.

Question 24

The odd natural numbers have been divided into groups as : $(1,3) ;(5,7,9,11) ;(13,15, \ldots, 23)$;...

show that the sum of numbers in the $n$th group is $4 n^{3}$.

Type 5

Question 25

Find the sum to $n$ terms of the following series :

(i) $2+5+14+41+\ldots$

(ii) $1+5+13+29+61+\ldots$

(iii) $1+4+13+40+121+\ldots$

(iv) $5+11+19+29+41+\ldots$

(v) $3+5+9+17+33+\ldots$