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 H_{1}, H_{2} 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

Explanation:

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

a) n

b) r

c) both n,r

d) All of the Above

Explanation: