If $\mathrm { a } , \mathrm { b } , \mathrm { c }$ are in A.P., $\mathrm { a } ^ { 2 } , \mathrm { b} ^ { 2 } , \mathrm { c } ^ { 2 }$ are in H.P., then prove that either $\mathrm { a } = \mathrm { b } = \mathrm { c }$ or $\mathrm { a } , \mathrm { b } , - \mathrm { c } / 2$ form a G.P.

Let $$S _ { n } = \sum _ { k = 1 } ^ { n } \frac { n } { n ^ { 2 } + k n + k ^ { 2 } } \quad \text { and } \quad T _ { n } = \sum _ { k = 0 } ^ { n - 1 } \frac { n } { n ^ { 2 } + k n + k ^ { 2 } } ,$$ for $n = 1,2,3 , \cdots$. Then,
(A) $\quad S _ { n } < \frac { \pi } { 3 \sqrt { 3 } }$
(B) $\quad S _ { n } > \frac { \pi } { 3 \sqrt { 3 } }$
(C) $T _ { n } < \frac { \pi } { 3 \sqrt { 3 } }$
(D) $T _ { n } > \frac { \pi } { 3 \sqrt { 3 } }$

If the sum of first $n$ terms of an A.P. is $cn^{2}$, then the sum of squares of these $n$ terms is
(A) $\frac{n\left(4n^{2}-1\right)c^{2}}{6}$
(B) $\frac{n\left(4n^{2}+1\right)c^{2}}{3}$
(C) $\frac{n\left(4n^{2}-1\right)c^{2}}{3}$
(D) $\frac{n\left(4n^{2}+1\right)c^{2}}{6}$

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots , a _ { 11 }$ be real numbers satisfying $\mathrm { a } _ { 1 } = 15 , \quad 27 - 2 \mathrm { a } _ { 2 } > 0$ and $\mathrm { a } _ { \mathrm { k } } = 2 \mathrm { a } _ { \mathrm { k } - 1 } - \mathrm { a } _ { \mathrm { k } - 2 }$ for $\mathrm { k } = 3,4 , \ldots , 11$.
If $\frac { a _ { 1 } ^ { 2 } + a _ { 2 } ^ { 2 } + \ldots + a _ { 11 } ^ { 2 } } { 11 } = 90$, then the value of $\frac { a _ { 1 } + a _ { 2 } + \ldots + a _ { 11 } } { 11 }$ is equal to

Let $a, b, c$ be positive integers such that $\frac{b}{a}$ is an integer. If $a, b, c$ are in geometric progression and the arithmetic mean of $a, b, c$ is $b + 2$, then the value of $$\frac{a^2 + a - 14}{a + 1}$$ is

Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6:11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is

Let $b _ { i } > 1$ for $i = 1,2 , \ldots , 101$. Suppose $\log _ { e } b _ { 1 } , \log _ { e } b _ { 2 } , \ldots , \log _ { e } b _ { 101 }$ are in Arithmetic Progression (A.P.) with the common difference $\log _ { e } 2$. Suppose $a _ { 1 } , a _ { 2 } , \ldots , a _ { 101 }$ are in A.P. such that $a _ { 1 } = b _ { 1 }$ and $a _ { 51 } = b _ { 51 }$. If $t = b _ { 1 } + b _ { 2 } + \cdots + b _ { 51 }$ and $s = a _ { 1 } + a _ { 2 } + \cdots + a _ { 51 }$, then
(A) $s > t$ and $a _ { 101 } > b _ { 101 }$
(B) $s > t$ and $a _ { 101 } < b _ { 101 }$
(C) $s < t$ and $a _ { 101 } > b _ { 101 }$
(D) $s < t$ and $a _ { 101 } < b _ { 101 }$

The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be a sequence of positive integers in arithmetic progression with common difference 2. Also, let $b _ { 1 } , b _ { 2 } , b _ { 3 } , \ldots$ be a sequence of positive integers in geometric progression with common ratio 2. If $a _ { 1 } = b _ { 1 } = c$, then the number of all possible values of $c$, for which the equality
$$2 \left( a _ { 1 } + a _ { 2 } + \cdots + a _ { n } \right) = b _ { 1 } + b _ { 2 } + \cdots + b _ { n }$$
holds for some positive integer $n$, is $\_\_\_\_$

Let $l _ { 1 } , l _ { 2 } , \ldots , l _ { 100 }$ be consecutive terms of an arithmetic progression with common difference $d _ { 1 }$, and let $w _ { 1 } , w _ { 2 } , \ldots , w _ { 100 }$ be consecutive terms of another arithmetic progression with common difference $d _ { 2 }$, where $d _ { 1 } d _ { 2 } = 10$. For each $i = 1,2 , \ldots , 100$, let $R _ { i }$ be a rectangle with length $l _ { i }$, width $w _ { i }$ and area $A _ { i }$. If $A _ { 51 } - A _ { 50 } = 1000$, then the value of $A _ { 100 } - A _ { 90 }$ is $\_\_\_\_$.

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be an arithmetic progression with $a _ { 1 } = 7$ and common difference 8. Let $T _ { 1 } , T _ { 2 } , T _ { 3 } , \ldots$ be such that $T _ { 1 } = 3$ and $T _ { n + 1 } - T _ { n } = a _ { n }$ for $n \geq 1$. Then, which of the following is/are TRUE ?
(A) $T _ { 20 } = 1604$
(B) $\sum _ { k = 1 } ^ { 20 } T _ { k } = 10510$
(C) $T _ { 30 } = 3454$
(D) $\sum _ { k = 1 } ^ { 30 } T _ { k } = 35610$

Let $7 \overbrace { 5 \cdots 5 } ^ { r } 7$ denote the $( r + 2 )$ digit number where the first and the last digits are 7 and the remaining $r$ digits are 5. Consider the sum $S = 77 + 757 + 7557 + \cdots + 7 \overbrace { 5 \cdots 5 } ^ { 98 } 7$. If $S = \frac { 7 \overbrace { 5 \cdots 5 } ^ { 99 } 7 + m } { n }$, where $m$ and $n$ are natural numbers less than 3000, then the value of $m + n$ is

Let $\mathbb { R }$ denote the set of all real numbers. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function such that $f ( x ) > 0$ for all $x \in \mathbb { R }$, and $f ( x + y ) = f ( x ) f ( y )$ for all $x , y \in \mathbb { R }$.
Let the real numbers $a _ { 1 } , a _ { 2 } , \ldots , a _ { 50 }$ be in an arithmetic progression. If $f \left( a _ { 31 } \right) = 64 f \left( a _ { 25 } \right)$, and
$$\sum _ { i = 1 } ^ { 50 } f \left( a _ { i } \right) = 3 \left( 2 ^ { 25 } + 1 \right)$$
then the value of
$$\sum _ { i = 6 } ^ { 30 } f \left( a _ { i } \right)$$
is $\_\_\_\_$ .

A man saves Rs. 200 in each of the first three months of his service. In each of the subsequent months his saving increases by Rs. 40 more than the saving of immediately previous month. His total saving from the start of service will be Rs. 11040 after
(1) 19 months
(2) 20 months
(3) 21 months
(4) 18 months

If $a, b, c, d$ and $p$ are distinct real numbers such that $\left(a^{2}+b^{2}+c^{2}\right)p^{2} - 2p(ab+bc+cd) + \left(b^{2}+c^{2}+d^{2}\right) \leq 0$, then
(1) $a, b, c, d$ are in A.P.
(2) $ab = cd$
(3) $ac = bd$
(4) $a, b, c, d$ are in G.P.

If the A.M. between $p ^ { \text {th} }$ and $q ^ { \text {th} }$ terms of an A.P. is equal to the A.M. between $r ^ { \text {th} }$ and $s ^ { \text {th} }$ terms of the same A.P., then $p + q$ is equal to
(1) $r + s - 1$
(2) $r + s - 2$
(3) $r + s + 1$
(4) $r + s$

The sum of the series $$\frac{1}{1+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + \frac{1}{\sqrt{3}+\sqrt{4}} + \ldots$$ upto 15 terms is
(1) 1
(2) 2
(3) 3
(4) 4

The sum of the series $1 ^ { 2 } + 2.2 ^ { 2 } + 3 ^ { 2 } + 2.4 ^ { 2 } + 5 ^ { 2 } + 2.6 ^ { 2 } + \ldots . + 2 ( 2 m ) ^ { 2 }$ is
(1) $m ( 2 m + 1 ) ^ { 2 }$
(2) $m ^ { 2 } ( m + 2 )$
(3) $m ^ { 2 } ( 2 m + 1 )$
(4) $m ( m + 2 ) ^ { 2 }$

The sum of the series $1 + \frac { 4 } { 3 } + \frac { 10 } { 9 } + \frac { 28 } { 27 } + \ldots$ upto $n$ terms is
(1) $\frac { 7 } { 6 } n + \frac { 1 } { 6 } - \frac { 2 } { 3.2 ^ { n - 1 } }$
(2) $\frac { 5 } { 3 } n - \frac { 7 } { 6 } + \frac { 1 } { 2.3 ^ { n - 1 } }$
(3) $n + \frac { 1 } { 2 } - \frac { 1 } { 2 \cdot 3 ^ { n } }$
(4) $n - \frac { 1 } { 3 } - \frac { 1 } { 3.2 ^ { n - 1 } }$

If the sum of the series $1 ^ { 2 } + 2 \cdot 2 ^ { 2 } + 3 ^ { 2 } + 2 \cdot 4 ^ { 2 } + 5 ^ { 2 } + \ldots 2.6 ^ { 2 } + \ldots$ upto n terms, when n is even, is $\frac { n ( n + 1 ) ^ { 2 } } { 2 }$, then the sum of the series, when n is odd, is
(1) $n ^ { 2 } ( n + 1 )$
(2) $\frac { n ^ { 2 } ( n - 1 ) } { 2 }$
(3) $\frac { n ^ { 2 } ( n + 1 ) } { 2 }$
(4) $n ^ { 2 } ( n - 1 )$

If $100$ times the $100^{\text{th}}$ term of an AP with non-zero common difference equals the $50$ times its $50^{\text{th}}$ term, then the $150^{\text{th}}$ term of this AP is
(1) $-150$
(2) 150 times its $50^{\text{th}}$ term
(3) 150
(4) zero

The sum of the series : $( 2 ) ^ { 2 } + 2 ( 4 ) ^ { 2 } + 3 ( 6 ) ^ { 2 } + \ldots$ upto 10 terms is :
(1) 11300
(2) 11200
(3) 12100
(4) 12300

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be an A.P, such that $\frac { a _ { 1 } + a _ { 2 } + \ldots + a _ { p } } { a _ { 1 } + a _ { 2 } + a _ { 3 } + \ldots + a _ { q } } = \frac { p ^ { 3 } } { q ^ { 3 } } ; p \neq q$. Then $\frac { a _ { 6 } } { a _ { 21 } }$ is equal to:
(1) $\frac { 41 } { 11 }$
(2) $\frac { 31 } { 121 }$
(3) $\frac { 11 } { 41 }$
(4) $\frac { 121 } { 1861 }$

If $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots , a _ { n } , \ldots$ are in A.P. such that $a _ { 4 } - a _ { 7 } + a _ { 10 } = m$, then the sum of first 13 terms of this A.P., is :
(1) 10 m
(2) 12 m
(3) 13 m
(4) 15 m

Given a sequence of 4 numbers, first three of which are in G.P. and the last three are in A.P. with common difference six. If first and last terms of this sequence are equal, then the last term is:
(1) 16
(2) 8
(3) 4
(4) 2