1. If \[S_{n}=\sum_{r=1}^{n} t_{r}=\frac{1}{6}n\left(n+1\right)\left(n+2\right) \forall n\geq 1\]
then
\[\lim_{n \rightarrow \infty}\sum_{r=1}^{n} \frac{1}{t_{r}}\] is
a) 1
b) 3/2
c) 2
d) 5/2
Explanation:
2. If unit’s digit of \[1^{2}+2^{2}+....+n^{2}\] is 5, then unit’s
digit of \[1^{10}+2^{10}+....+n^{10}\] is
a) 1
b) 3
c) 5
d) 7
Explanation: Use the fact that n10 and n2 have the same unit’s digit
3. Four geometric means are inserted between \[2^{11}-1\]
and \[2^{11}+1\] . The product of these geometric means
is
a) \[2^{44}-1\]
b) \[2^{44}-2^{23}+1\]
c) \[2^{44}-2^{22}+1\]
d) \[2^{45}-1\]
Explanation:
4. Let \[S_{n}=\frac{1}{2^{2}-1}+\frac{1}{4^{2}-1}+....+\frac{1}{\left(2n\right)^{2}-1}\] then value
of \[S_{25}\] is
a) 25/54
b) 25/53
c) 25/51
d) 1/2
Explanation:
5. Suppose \[t_{r}=1^{2}+2^{2}+....+r^{2}\] . Let
\[t_{1}+t_{2}+....+t_{n}=\frac{1}{12}n\left(n+1\right)\left(n+2\right)k\]
then value of k is
a) n+1
b) 2n+1
c) 3n-1
d) n
Explanation:
6. Let a, b, c > 0 and a, b, c be in A.P. If \[a^{2},b^{2},c^{2}\] are
in G.P., then
a) a = b = c
b) \[a^{2}+b^{2}=c^{2}\]
c) \[a^{2}+c^{2}=3b^{2}\]
d) none of these
Explanation: Use a, b, c are in G.P
7. Sum of the infinite series \[\frac{1}{3}+\frac{3}{\left(3\right)\left(7\right)}+\frac{5}{\left(3\right)\left(7\right)\left(11\right)}+\frac{7}{\left(3\right)\left(7\right)\left(11\right)\left(15\right)}+....\]
is
a) 1/2
b) 1/3
c) 1/6
d) 1/4
Explanation:
8. Let S = \[\frac{4}{19}+\frac{44}{19^{2}}+\frac{444}{19^{3}}+.... \infty\]
Then S is equal to
a) 40/9
b) 38/81
c) 36/171
d) none of these
Explanation:
9. If a, b, c, ... are in G.P., and \[a^{1/x}=b^{1/y}=c^{1/z}...,\] , then
x, y, z, ... are in
a) H.P
b) G.P
c) A.P
d) none of these
Explanation:
10. If \[a^{x}=b^{y}=c^{z}=d^{u}\] and a, b, c, d are in G.P., then
x, y, z, u are in
a) A.P
b) G.P
c) H.P
d) none of these
Explanation: H.P