## Sequence and Series Questions and Answers Part-9

1. If $S_{n}=\sum_{r=1}^{n} t_{r}=\frac{1}{6}n\left(n+1\right)\left(n+2\right) \forall n\geq 1$
then $\lim_{n \rightarrow \infty}\sum_{r=1}^{n} \frac{1}{t_{r}}$    is
a) 1
b) 3/2
c) 2
d) 5/2

Explanation:

2. If unit’s digit of $1^{2}+2^{2}+....+n^{2}$    is 5, then unit’s digit of $1^{10}+2^{10}+....+n^{10}$     is
a) 1
b) 3
c) 5
d) 7

Explanation: Use the fact that n10 and n2 have the same unit’s digit

3. Four geometric means are inserted between $2^{11}-1$   and $2^{11}+1$   . The product of these geometric means is
a) $2^{44}-1$
b) $2^{44}-2^{23}+1$
c) $2^{44}-2^{22}+1$
d) $2^{45}-1$

Explanation:

4. Let $S_{n}=\frac{1}{2^{2}-1}+\frac{1}{4^{2}-1}+....+\frac{1}{\left(2n\right)^{2}-1}$         then value of $S_{25}$ is
a) 25/54
b) 25/53
c) 25/51
d) 1/2

Explanation:

5. Suppose $t_{r}=1^{2}+2^{2}+....+r^{2}$     . Let
$t_{1}+t_{2}+....+t_{n}=\frac{1}{12}n\left(n+1\right)\left(n+2\right)k$
then value of k is
a) n+1
b) 2n+1
c) 3n-1
d) n

Explanation:

6. Let a, b, c > 0 and a, b, c be in A.P. If $a^{2},b^{2},c^{2}$  are in G.P., then
a) a = b = c
b) $a^{2}+b^{2}=c^{2}$
c) $a^{2}+c^{2}=3b^{2}$
d) none of these

Explanation: Use a, b, c are in G.P

7. Sum of the infinite series $\frac{1}{3}+\frac{3}{\left(3\right)\left(7\right)}+\frac{5}{\left(3\right)\left(7\right)\left(11\right)}+\frac{7}{\left(3\right)\left(7\right)\left(11\right)\left(15\right)}+....$
is
a) 1/2
b) 1/3
c) 1/6
d) 1/4

Explanation:

8. Let S = $\frac{4}{19}+\frac{44}{19^{2}}+\frac{444}{19^{3}}+.... \infty$
Then S is equal to
a) 40/9
b) 38/81
c) 36/171
d) none of these

Explanation:

9. If a, b, c, ... are in G.P., and $a^{1/x}=b^{1/y}=c^{1/z}...,$    , then x, y, z, ... are in
a) H.P
b) G.P
c) A.P
d) none of these

10. If $a^{x}=b^{y}=c^{z}=d^{u}$     and a, b, c, d are in G.P., then x, y, z, u are in