Sequence and Series Questions and Answers Part-8

1. If $a_{1},a_{2},...,a_{n}$    are in H.P., then
$\frac{a_{1}}{a_{2}+a_{3}+....+a_{n}},\frac{a_{2}}{a_{1}+a_{3}+....+a_{n}},....\frac{a_{n}}{a_{1}+a_{2}+....+a_{n-1}}$
are in
a) A.P
b) H.P
c) G.P
d) A.G.P.

Explanation:

2.Let $S_{n}=\sum_{r=1}^{n}t_{r}=\frac{1}{6}n\left(2n^{2}+9n+13\right) ,then \sum_{r=1}^{n}\sqrt{t_{r}}$
equals
a) $\frac{1}{2}n\left(n+1\right)$
b) $\frac{1}{2}n\left(n+3\right)$
c) $\left(n+1\right)^{2}$
d) $n^{2}$

Explanation:

3. Let $a_{n}=\underbrace{111.......1}_{n }$   . The remainder when $a_{124}$  is divided by 271 is
a) 23
b) 25
c) 27
d) 29

Explanation:

4. Three arithmetic means, three geometric means and three harmonic means are inserted between 1 and 5. The cubic equation whose roots are 3rd A.M., 2nd G.M. and 1st H.M. is
a) $x^{3}-\frac{1}{4}\left(21+4\sqrt{5}\right)x^{2}+5x-5\sqrt{5}=0$
b) $x^{3}-\frac{1}{4}\left(21+4\sqrt{5}\right)x^{2}+\left(5+\frac{21}{4}\sqrt{5}\right)x-5\sqrt{5}=0$
c) $4x^{3}-\left(21+4\sqrt{5}\right)x^{2}+\left(15+21\sqrt{5}\right)-20\sqrt{5}=0$
d) none of these

Explanation:

5. If $t_{r}$ denotes the rth term of an A.P., and tp = 1/q, $t_{q}=1/p$   then which of the following is a root of the equation $\left(p + 2q + 3r\right) x^{2} + \left(q + 2r – 3p\right) x +\left(r + 2p – 3q\right) = 0$
a) $t_{p}$
b) $t_{q}$
c) $t_{pq}$
d) $t_{p+q}$

Explanation: Clearly 1 is a root of the equation and tpq = 1

6. If the pth term of an A.P. is 1/q and qth term is 1/p and sum of pq terms is 25pq, then p and q are connected by
a) $p^{2}=4q^{2}+1$
b) p = 4q – 1
c) pq = 4 + p
d) pq = 25

Explanation:

7. If $p\left(x\right)=\frac{1+x^{2}+x^{4}+...+x^{2n-2}}{1+x+x^{2}+....+x^{n-1}}$
is a polynomial in x, then n must be
a) odd
b) even
c) greater than 5
d) less than 5

Explanation:

8. $\sqrt{\underbrace{111......1}_{200 digit }-\underbrace{222.....2}_{100 digit }}$        equals
a) $\sqrt{\underbrace{1313......13}_{100 digit }}$
b) $\sqrt{\underbrace{33......3}_{100 digit }}$
c) $\sqrt{\underbrace{2323......23}_{100 digit }}$
d) none of these

Explanation:

9. Let $S_{n}\left(a\right)=1+2\left(1-a\right)+3\left(1-a\right)\left(1-2a\right)+4\left(1-a\right)\left(1-2a\right)\left(1-3a\right)+....$
upto n terms, then $S_{100}\left(\frac{1}{99}\right)$  equals
a) 100/99
b) 1/99
c) 200
d) 99

10. Let $a_{n}=\int_{0}^{\pi/2} \frac{1-\cos2n x}{1-\cos2x}dx$       then $a_{1},a_{2},a_{3}...$   are in