## Sequence and Series Questions and Answers Part-5

1. Value of $\lim_{n \rightarrow \infty}\sum_{r=1}^{n}\tan^{-1}\left(\frac{1}{2r^{2}}\right)$
is
a) $\pi/2$
b) $\pi/4$
c) 1
d) $\pi/8$

Explanation:

2. Let $I_{n}=\int_{0}^{\pi/4} \tan^{n}x dx$     . then $I_{2}+I_{4},I_{3}+I_{5},I_{4}+I_{6},I_{5}+I_{7},....$
are in
a) A.P.
b) G.P.
c) H.P.
d) none of these

Explanation: We have for r ≥ 1,

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

Explanation: We have, for n ≥ 1,

4. If $H_{n}=1+\frac{1}{2}+...+\frac{1}{n}$     , then value of $S_{n}=1+\frac{3}{2}+\frac{5}{3}+...+\frac{2n-1}{n}$      is
a) $H_{n}+n$
b) $2n-H_{n}$
c) $\left(n-1\right)+H_{n}$
d) $H_{n}+2n$

Explanation:

5. Sum to n terms of $S_{n}=1+\frac{3}{2}+\frac{7}{4}+\frac{15}{8}+\frac{31}{16}+....$
is
a) $2\left(n-1\right)+\frac{1}{2^{n-1}}$
b) $2n-\frac{1}{2^{n}}$
c) $2+\frac{1}{2^{n}}$
d) $2n-1+\frac{1}{2^{n}}$

Explanation:

6. Let $a_{k}=\frac{\sqrt{\left(k-1\right)k}}{\sqrt{2k-1+2\sqrt{k\left(k-1\right)}}}$
$k\epsilon N$   ,then
a) $a_{k}<\frac{1}{2}\sqrt{k} \forall k\epsilon N$
b) $\sum_{k=1}^{n}a_{k}<\frac{1}{2}\int_{0}^{n} \sqrt{x} dx$
c) $\sum_{k=1}^{n}a_{k}<\frac{1}{3}n \sqrt{n}$
d) All of the Above

Explanation:

7. For 0 < x < 1, x > 1, let $a_{k}= \frac{kx^{k-1}}{1+x+...+x^{k-1}}$
then $\sum_{k=1}^{n}a_{k}$   connot exceed
a) $\frac{\left(\sqrt{x}\right)^{n}-1}{\sqrt{x}-1}$
b) $\frac{2\left(x^{n}-1\right)}{x-1}$
c) $2\left(1+x+..+x^{n-1}\right)$
d) All of the Above

Explanation:

8. If a, b, c are in H.P., then
a) $\frac{a}{b+c-a},\frac{b}{c+a-b},\frac{c}{a+b-c}$       are in H.P
b) $\frac{2}{b}=\frac{1}{b-a}+\frac{1}{b-c}$
c) $a-\frac{b}{2},\frac{b}{2},c-\frac{b}{2}$    are in G.P
d) All of the Above

Explanation:

9. Let $a_{n}=2n+5,b_{n}=\left(\frac{1}{3}\right)^{n}\forall n\epsilon N$      . If $\sum_{k=1}^{n}a_{k}b_{k}c_{k}=\frac{1}{3}\left(n+1\right)\left(n+2\right)\left(n+3\right)$        , then
a) $c_{1}=24/7$
b) $c_{n}=\frac{\left(n+1\right)\left(n+2\right)3^{n}}{2n+5} \forall n \geq 2$
c) $\sum_{n=1}^{\infty}\frac{1}{c_{n}}=\frac{29}{72}$
d) All of the Above

10.Let $a_{1}=1 , a_{n+1}=\frac{1}{2}a_{n}+\frac{n^{2}-2n-1}{n^{2}\left(n+1\right)^{2}} \forall n\geq 1$
and $S_{n}=a_{1}+a_{2}+...+a_{n}$
a) $a_{n}=\frac{2}{n^{2}}-\left(\frac{1}{2}\right)^{n-1}\forall n\geq 1$
b) $S_{n}$ is minimum for n = 3
c) $S_{n}$ is minimum for n = 4