## Sequence and Series Questions and Answers Part-10

1. Let $F_{0}=1,F_{1}=1$   and $F_{n+1}=F_{n}+F_{n-1}\forall n\geq 1$
Sum of the series $\sum_{n=1}^{\infty}\frac{F_{n}}{F_{n-1}F_{n+1}}$    is
a) 1
b) 2
c) 1/2
d) 2/3

Explanation:

2. If the sum to n terms of an A.P. is $3n^{2}+5n$  , while $T_{m}=164$  , then value of m is
a) 25
b) 26
c) 27
d) 28

Explanation:

3. If $G_{1}$ and $G_{2}$ are two geometric means and A is the arithmetic mean inserted between two positive numbers a and b then the value of $\frac{G_1^2}{G_{2}}+\frac{G_2^2}{G_{1}}$   is
a) A
b) 2A
c) A/2
d) 3A/2

Explanation:

4. If $A_{1},A_{2}$  be two arithmetic means and $G_{1},G_{2}$  be two geometric means between two positive numbers a and b, then $\frac{A_{1}+A_{2}}{G_{1}G_{2}}$    is equal to
a) $\frac{a}{b}+\frac{b}{a}$
b) $\frac{1}{a}+\frac{1}{b}$
c) $\sqrt{\frac{a}{b}+\frac{b}{a}}$
d) $\frac{ab}{a+b}$

Explanation:

5. Suppose for each $n\epsilon N$ .
$\left(1^{2}-a_{1}\right)\left(2^{2}-a_{2}\right)+....+\left(n^{2}-a_{n}\right)=\frac{1}{3}n\left(n^{2}-1\right)$
then $a_{n}$ equals
a) n
b) n-1
c) n+1
d) 2n

Explanation:

6. Let $A_{1},A_{2}$  be two arithmetic means, $G_{1},G_{2}$   be two geometric means, and H1, H2 be two harmonic means between two positive numbers a and b. The value of $\frac{G_{1}G_{2}}{H_{1}H_{2}}.\frac{H_{1}+H_{2}}{A_{1}+A_{2}}$     is
a) 1/2
b) 1
c) 3/2
d) 2

Explanation:

7. Sum of the series S = (n) (n) + (n – 1) (n + 1) + (n – 2) (n + 2) + ... + 1(2n + 1) is
a) $n^{3}$
b) $\frac{1}{6}n\left(n+1\right)\left(n+2\right)$
c) $\frac{1}{3}n^{3}-n^{2}$
d) none of these

Explanation:

8. Let $a,d \epsilon \left(0,\infty\right)$   and $a_{r}=a+\left(r-1\right)d\forall r\epsilon N$
If $S_{k}=\sum_{i=1}^{k}\frac{1}{a_{i}}$     then $\sum_{k=1}^{n}\frac{k}{S_{k}}$     cannot exceed
a) $\frac{n}{4}\left(3a_{1}+a_{n}\right)$
b) $n\left(3a_{1}+a_{n}\right)$
c) $\sum_{k=1}^{n}a_{k}$
d) All of the Above

Explanation:

9. Let $S_{n}=\frac{3}{2}.\frac{1}{1^{2}}+\frac{5}{2}.\frac{1+2}{1^{2}+2^{2}}+\frac{7}{2}.\frac{1+2+3}{1^{2}+2^{2}+3^{2}}+....$
upto n terms, then $S_{n}$ cannot exceed
a) 4n
b) 2n
c) 3n
d) All of the Above

10. Let x and y be two positive real numbers. Let P be the rth mean when n arithmetic means are inserted between x and y and Q be the rth harmonic mean between x and y when n harmonic means are inserted between x and y, then $\frac{P}{x}+\frac{y}{Q}$   is independent of