## Sequence and Series Questions and Answers Part-1

1. Let $a_{k}=\frac{\left(k^{2}+1\right)^{2}}{k^{2}+4} ,k \epsilon N$    , then value of $a_1^8a_2^7....a_7^2a_{8}$    is equal to
a) $\frac{256}{41}$
b) $\frac{128}{41}$
c) $\frac{64}{41}$
d) $\frac{32}{41}$

Explanation:

2. Suppose a, b, c are positive real numbers and that a, b, c are pth, qth, rth terms of a G.P., then least value of $E=a^{q-r}+b^{r-p}+c^{p-q}$     is
a) 3
b) 6
c) 9
d) 12

Explanation: As A.M. ≥ G.M

3. Let $a_{2},a_{2},....a_{n}>0$    be such that $\sum_{k=1}^{n}a_{k}=\frac{1}{2}n\left(n+1\right)$     , then least value of $\sum_{k=1}^{n}\frac{\left(k^{2}-1\right)a_{k}+k^{2}+2k}{a_k^2+a_{k}+1}$
is
a) $\frac{1}{2}n^{2}$
b) $\frac{1}{2}n\left(n+1\right)$
c) 2n
d) $n^{2}-1$

Explanation:

4. Suppose $a,d \epsilon \left(0,\infty\right)$   and $a_{n}=a+\left(n-1\right)d\forall n\epsilon N$       . Least value of $\sum_{k=1}^{n}\sqrt{a_k^2-a_{k}a_{k+1}+a_{k+1}^2 }$
is
a) $\frac{n}{n+1}\sum_{k=1}^{n+1}a_{k}$
b) $\frac{1}{n+1}\sum_{k=1}^{n}a_{k}$
c) $\frac{n}{n+1}\sum_{k=1}^{n}a_{k}$
d) $\frac{n}{n+1}\sum_{k=2}^{n}a_{k}$

Explanation: For a, b > 0,

5. Let S denote the set of all real values of x for which $\left(x^{2020}+1\right)\left(1+x^{2}+x^{4}+....+x^{2018}\right)=2020x^{2019}$
then the number of elements in S is
a) 0
b) 1
c) 2
d) infinite

Explanation:

6. Let $a_{n}=\sum_{k=1}^{n}\frac{1}{k\left(n+1-k\right)}$     , then for $n\geq 2$
a) $a_{n+1}> a_{n}$
b) $a_{n+1}< a_{n}$
c) $a_{n+1}= a_{n}$
d) $a_{n+1}-a_{n}=1/n$

Explanation:

7. If a, b, c are three unequal numbers such that a, b, c are in A.P. and b – a, c – b, a are in G.P., then ratio a : b : c is equal
a) 1 : 2 : 3
b) 1 : 3 : 4
c) 2 : 3 : 4
d) 1 : 2 : 4

Explanation: By the hypothesis, b – a = c – b and

8. If 1, $\log _{y}x,\log _{z}y,-15\log _{x}z$     are in A.P., then
a) $z^{3}=x$
b) $x=y^{-2}$
c) $z^{-2}=y$
d) x=y

Explanation: Let d be the common difference. Then

9. If $\log 2,\log \left(2^{x}-1\right)$   and $\log \left(2^{x}+3\right)$   are in A.P., then x is equal to
a) $\frac{5}{2}$
b) $\log _{2}5$
c) $\log _{3}2$
d) $\frac{3}{2}$

10. The sum $S_{n}$ to n terms of the series $\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+....$     is equal to
a) $2^{n}-n-1$
b) $1-2^{-n}$
c) $2^{-n}+n-1$
d) $2^{n}-1$