## Sequence and Series Questions and Answers Part-2

1. If $1^{2}+2^{2}+3^{2}+...+2003^{2}=\left(2003\right)\left(4007\right)\left(334\right)$
and $\left(1\right)\left(2003\right)+\left(2\right)\left(2002\right)+\left(3\right)\left(2001\right)+...+\left(2003\right)\left(1\right)=\left(2003\right)\left(334\right)\left(x\right)$
then x equals
a) 2005
b) 2004
c) 2003
d) 2001

Explanation:

2. Sum of the series $\frac{1}{2^{2}-1}+\frac{1}{4^{2}-1}+\frac{1}{6^{2}-1}+....$       upto 2n terms is
a) $\frac{n}{2n+1}$
b) $\frac{1}{2n+1}$
c) $\frac{1}{n+1}$
d) $\frac{4}{2n-1}$

Explanation:

3. Let $a_{1},a_{2},....,a_{10}$   be in A.P. and $h_{1},h_{2},....,h_{10}$   be in H.P. If $a_{1}=h_{1}=2$   and $a_{10}=h_{10}=3$   , then a5 h6 is
a) 2
b) 3
c) 5
d) 6

Explanation: Let d be the common difference of the A.P., then

4. Suppose a, b, c are in A.P. and $a^{2},b^{2},c^{2}$   are in G.P. If a < b < c and a + b + c = 3/2, then the value of a is
a) $\frac{1}{2\sqrt{2}}$
b) $\frac{1}{2\sqrt{3}}$
c) $\frac{1}{2}-\frac{1}{\sqrt{3}}$
d) $\frac{1}{2}-\frac{1}{\sqrt{2}}$

Explanation:

5. Let $S_{1},S_{2},....$   be squares such that for each n ≥ 1, the length of a side of $S_{n}$ equals the length of a diagonal of $S_{n+1}$ . If the length of a side of $S_{1}$ is 10 cm, then the smallest value of n for which Area $\left(S_{n}\right)< 1$   is
a) 7
b) 8
c) 9
d) 10

Explanation: Let an denote the length of a side of Sn. We are given:

6. If a, b, c are in G.P., and log a – log 2b, log 2b – log 3c and log 3c – log a are in A.P., then a, b, c are the lengths of the sides of a triangle which is
a) acute-angled
b) obtuse-angled
c) right-angled
d) equilateral.

Explanation: We have b2= ac and

7. If $x_{1},x_{2}...x_{n}$   are n non-zero real numbers such that $\left(x_2^1+x_2^2+...+x_{n-1}^{2}\right)\left(x_2^2+x_3^2+...+x_{n}^{2}\right)\leq \left(x_{1}x_{2}+x_{2}x_{3}+...+x_{n-1}x_{n}\right)^{2}$
then $x_{1},x_{2},...,x_{n}$   are in
a) A.P
b) G.P
c) H.P
d) A.G.P

Explanation: We shall make use of the identity

8. If three positive real numbers a, b, c (c > a) are in H.P., then log (a + c) + log (a – 2b + c) is equal to
a) 2 log (c – b)
b) 2 log (a + c)
c) 2 log (c – a)
d) log a + log b + log c.

Explanation: a, b, c are in H.P., b = (2ac)/(a + c). We have

9. If $a_{1},a_{2},...,a_{n}$   are in A.P. with common difference $d\neq 0$ , then the sum of the series
sin d [cosec a1 cosec a2 + cosec a2 cosec a3+....+cosec an-1cosec an] is
a) $\sec a_{1}-\sec a_{n}$
b) cosec a1 - cosec an
c) $\cot a_{1}-\cot a_{n}$
d) $\tan a_{1}-\tan a_{n}$

10. If sum of first n terms of the series $S=\sqrt{3}+\sqrt{75}+\sqrt{243}+\sqrt{507}+....$       is $435\sqrt{3}$ , then n is equal to