## Sequence and Series Questions and Answers Part-3

1. Sum to n terms of the series $\frac{1}{1.2.3.4}+\frac{1}{2.3.4.5}+\frac{1}{3.4.5.6}+....$
is
a) $\frac{n^{3}}{3\left(n+1\right)\left(n+2\right)\left(n+3\right)}$
b) $\frac{n^{3}+6n^{2}-3n}{6\left(n+2\right)\left(n+3\right)\left(n+4\right)}$
c) $\frac{15n^{2}+7n}{4n\left(n+1\right)\left(n+5\right)}$
d) $\frac{n^{3}+6n^{2}+11n}{18\left(n+1\right)\left(n+2\right)\left(n+3\right)}$

Explanation: The rth term of the series is given by

2. Sum of all the products of the first n positive integers taken two at a time is
a) $\frac{1}{24}\left(n-1\right)n\left(n+1\right)\left(3n+2\right)$
b) $\frac{1}{48}\left(n-2\right)\left(n-1\right)n^{2}$
c) $\frac{1}{6}n\left(n+1\right)\left(n+2\right)\left(n+5\right)$
d) $\frac{1}{3}n\left(n+1\right)\left(n+2\right)$

Explanation:

3. For a positive integer n, let $a\left(n\right)=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+....+\frac{1}{\left(2^{n}\right)-1}$
a) a(100) < 100
b) a(100) > 100
c) a(200) < 100
d) a(2000) > 1000

Explanation:

4. $S_{n}$ , the sum to n terms of the series $\left(n^{2}-1^{2}\right)+2\left(n^{2}-2^{2}\right)+3\left(n^{2}-3^{2}\right)+....$
is
a) $\frac{1}{4}n^{2}\left(n^{2}-1\right)$
b) $\frac{1}{4}n\left(n+1\right)^{2}$
c) 0
d) $2n\left(n^{2}-1\right)$

Explanation:

5. Let x be the arithmetic mean and y, z be the two geometric means between any two positive numbers. Then value of $\frac{y^{3}+z^{3}}{xyz}$   is
a) 2
b) 3
c) 1/2
d) 3/2

Explanation: Let two positive numbers be a and b. Then

6. Let $H_{n}=1+\frac{1}{2}+....+\frac{1}{n}$    , then the sum to n terms of the series $\frac{1^{2}}{1^{3}}+\frac{1^{2}+2^{2}}{1^{3}+2^{3}}+\frac{1^{2}+2^{2}+3^{2}}{1^{3}+2^{3}+3^{3}}+....$
is
a) $\frac{4}{3}H_{n}-1$
b) $\frac{4}{3}H_{n}+\frac{1}{n}$
c) $\frac{4}{3}H_{n}$
d) $\frac{4}{3}H_{n}-\frac{2}{3}\frac{n}{n+1}$

Explanation:

7. If the sum of first n terms of an A.P. is $cn^{2}$, then sum of squares of these n terms is
a) $\frac{1}{6}n\left(4n^{2}-1\right)c^{2}$
b) $\frac{1}{3}n\left(4n^{2}+1\right)c^{2}$
c) $\frac{1}{3}n\left(4n^{2}-1\right)c^{2}$
d) $\frac{1}{6}n\left(4n^{2}+1\right)c^{2}$

Explanation: Let tn denote the nth term of the the A.P., then

8. The sum of first n terms of the series $1^{2}+2\times 2^{2}+3^{2}+2 \times4^{2}+5^{2}+2\times 6^{2}+....$
is $n\left(n+1\right)^{2}/2$    when n is even. When n is odd the sum of the series is
a) $n^{2}\left(3n+1\right)/4$
b) $n^{2}\left(n+1\right)/2$
c) $n^{3}\left(n-1 \right)/2$
d) $n\left(n+2\right)/2$

Explanation: Let n = 2m, then

9. The interior angles of a polygon are in A.P. If the smallest angle is 120° and the common difference is 5°, then number of sides in the polygon is.
a) 7
b) 8
c) 9
d) 16

10. Let $A_{n}=\frac{1.2.3+2.3.4+3.4.5+.... + n }{n\left(1.2+2.3+3.4+...+ n \right)}$
then $\lim_{n \rightarrow \infty} A_{n}$   is
a) $\frac{3}{4}$
b) $\frac{1}{4}$
c) $\frac{1}{2}$
d) $\frac{5}{4}$