In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals
(1) $\frac { 1 } { 2 } ( 1 - \sqrt { 5 } )$
(2) $\frac { 1 } { 2 } \sqrt { 5 }$
(3) $\sqrt { 5 }$
(4) $\frac { 1 } { 2 } ( \sqrt { 5 } - 1 )$

The difference between the fourth term and the first term of a Geometrical Progression is 52. If the sum of its first three terms is 26, then the sum of the first six terms of the progression is
(1) 63
(2) 189
(3) 728
(4) 364

The sum of first 20 terms of the sequence $0.7, 0.77, 0.777, \ldots$ is
(1) $\frac{7}{81}(179-10^{-20})$
(2) $\frac{7}{9}(99-10^{-20})$
(3) $\frac{7}{81}(179+10^{-20})$
(4) $\frac{7}{9}(99+10^{-20})$

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

Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is:
(1) $2 - \sqrt { 3 }$
(2) $2 + \sqrt { 3 }$
(3) $\sqrt { 2 } + \sqrt { 3 }$
(4) $3 + \sqrt { 2 }$

If $m$ is the A.M. of two distinct real numbers $l$ and $n$ $(l, n > 1)$ and $G_1, G_2$ and $G_3$ are three geometric means between $l$ and $n$, then $G_1^4 + 2G_2^4 + G_3^4$ equals:
(1) $4l^2 m n$
(2) $4lm^2 n$
(3) $4lmn^2$
(4) $4l^2 m^2 n^2$

If $m$ is the A.M. of two distinct real numbers $l$ and $n$ $(l, n > 1)$ and $G _ { 1 } , G _ { 2 }$ and $G _ { 3 }$ are three geometric means between $l$ and $n$, then $G _ { 1 } ^ { 4 } + 2 G _ { 2 } ^ { 4 } + G _ { 3 } ^ { 4 }$ equals
(1) $4 l ^ { 2 } m ^ { 2 } n ^ { 2 }$
(2) $4 l ^ { 2 } m n$
(3) $4 l m ^ { 2 } n$
(4) $4 l m n ^ { 2 }$

If $b$ is the first term of an infinite G.P whose sum is five, then $b$ lies in the interval.
(1) $( - \infty , - 10 )$
(2) $( 10 , \infty )$
(3) $( 0,10 )$
(4) $( - 10,0 )$

If $b$ is the first term of an infinite geometric progression whose sum is five, then $b$ lies in the interval
(1) $[ 10 , \infty )$
(2) $( - \infty , - 10 ]$
(3) $( - 10,0 )$
(4) $( 0,10 )$

The sum of the first 20 terms of the series $1 + \frac { 3 } { 2 } + \frac { 7 } { 4 } + \frac { 15 } { 8 } + \frac { 31 } { 16 } + \ldots$ is
(1) $39 + \frac { 1 } { 2 ^ { 19 } }$
(2) $38 + \frac { 1 } { 2 ^ { 20 } }$
(3) $38 + \frac { 1 } { 2 ^ { 19 } }$
(4) $39 + \frac { 1 } { 2 ^ { 20 } }$

If $\alpha , \beta$ and $\gamma$ are three consecutive terms of a non-constant G.P. Such that the equations $\alpha x ^ { 2 } + 2 \beta x + \gamma = 0$ and $x ^ { 2 } + x - 1 = 0$ have a common root, then $\alpha ( \beta + \gamma )$ is equal to:
(1) $\beta \gamma$
(2) $\alpha \beta$
(3) $\alpha \gamma$
(4) 0

The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P. Then the sum of the original three terms of the G.P. is:

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$, be a G.P. such that $a _ { 1 } < 0 , a _ { 1 } + a _ { 2 } = 4$ and $a _ { 3 } + a _ { 4 } = 16$. If $\sum _ { i = 1 } ^ { 9 } a _ { i } = 4 \lambda$, then $\lambda$, is equal to.
(1) - 513
(2) - 171
(3) 171
(4) $\frac { 511 } { 3 }$

The sum of the first three terms of G.P is $S$ and their products is 27. Then all such $S$ lie in
(1) $(-\infty, -9] \cup [3, \infty)$
(2) $[-3, \infty)$
(3) $(-\infty, -3] \cup [9, \infty)$
(4) $(-\infty, 9]$

If $2 ^ { 10 } + 2 ^ { 9 } \cdot 3 ^ { 1 } + 2 ^ { 8 } \cdot 3 ^ { 2 } + \ldots\ldots + 2 \cdot 3 ^ { 9 } + 3 ^ { 10 } = S - 2 ^ { 11 }$, then $S$ is equal to
(1) $3 ^ { 11 } - 2 ^ { 12 }$
(2) $3 ^ { 11 }$
(3) $\frac { 3 ^ { 11 } } { 2 } + 2 ^ { 10 }$
(4) $2.3 ^ { 11 }$

Let $a _ { n }$ be the $n ^ { \text {th } }$ term of a G.P. of positive terms. If $\sum _ { n = 1 } ^ { 100 } a _ { 2 n + 1 } = 200$ and $\sum _ { n = 1 } ^ { 100 } a _ { 2 n } = 100$, then $\sum _ { n = 1 } ^ { 200 } a _ { n }$ is equal to:
(1) 300
(2) 225
(3) 175
(4) 150

The product $2 ^ { \frac { 1 } { 4 } } \bullet 4 ^ { \frac { 1 } { 16 } } \bullet 8 ^ { \frac { 1 } { 48 } } \bullet 16 ^ { \frac { 1 } { 128 } } \bullet \ldots$ to $\infty$ is equal to:
(1) $2 ^ { \frac { 1 } { 2 } }$
(2) $2 ^ { \frac { 1 } { 4 } }$
(3) 1
(4) 2

If $|x| < 1, |y| < 1$ and $x \neq 1$, then the sum to infinity of the following series $(x + y) + (x^{2} + xy + y^{2}) + (x^{3} + x^{2}y + xy^{2} + y^{3}) + \ldots$ is
(1) $\frac{x + y - xy}{(1 + x)(1 + y)}$
(2) $\frac{x + y + xy}{(1 + x)(1 + y)}$
(3) $\frac{x + y - xy}{(1 - x)(1 - y)}$
(4) $\frac{x + y + xy}{(1 - x)(1 - y)}$

If the sum of the second, third and fourth terms of a positive term G.P. is 3 and the sum of its sixth, seventh and eighth terms is 243, then the sum of the first 50 terms of this G.P. is:
(1) $\frac{1}{26}\left(3^{49}-1\right)$
(2) $\frac{1}{26}\left(3^{50}-1\right)$
(3) $\frac{2}{13}\left(3^{50}-1\right)$
(4) $\frac{1}{13}\left(3^{50}-1\right)$

Let $a , b , c , d$ and $p$ be non-zero distinct real numbers such that $\left( a ^ { 2 } + b ^ { 2 } + c ^ { 2 } \right) p ^ { 2 } - 2 ( a b + b c + c d ) p + \left( b ^ { 2 } + c ^ { 2 } + d ^ { 2 } \right) = 0$. Then
(1) $a , b , c$ are in A.P.
(2) $a , c , p$ are in G.P.
(3) $a , b , c , d$ are in G.P.
(4) $a , b , c , d$ are in A.P.

If $x = \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \tan ^ { 2 } \theta$ and $y = \sum _ { n = 0 } ^ { \infty } \cos ^ { 2 n } \theta$, for $0 < \theta < \frac { \pi } { 4 }$, then:
(1) $x ( 1 + y ) = 1$
(2) $y ( 1 - x ) = 1$
(3) $y ( 1 + x ) = 1$
(4) $x ( 1 - y ) = 1$

Let $S$ be the sum of the first 9 term of the series : $\{ x + k a \} + \left\{ x ^ { 2 } + ( k + 2 ) a \right\} + \left\{ x ^ { 3 } + ( k + 4 ) a \right\} + \left\{ x ^ { 4 } + ( k + 6 ) a \right\} + \ldots$ where $a \neq 0$ and $x \neq 1$. If $S = \frac { x ^ { 10 } - x + 45 a ( x - 1 ) } { x - 1 }$, then $k$ is equal to
(1) - 5
(2) 1
(3) - 3
(4) 3

If $f ( x + y ) = f ( x ) f ( y )$ and $\sum_{x=1}^{n} f(x) = 2$, then the value of $\sum_{x=1}^{n} f(x)$ is given. [Content truncated in source]

Three numbers are in an increasing geometric progression with common ratio $r$. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference $d$. If the fourth term of GP is $3 r ^ { 2 }$, then $r ^ { 2 } - d$ is equal to :
(1) $7 - \sqrt { 3 }$
(2) $7 + 3 \sqrt { 3 }$
(3) $7 - 7 \sqrt { 3 }$
(4) $7 + \sqrt { 3 }$

If the sum of an infinite GP, $a , a r , a r ^ { 2 } , a r ^ { 3 } , \ldots$ is 15 and the sum of the squares of its each term is 150 , then the sum of $a r ^ { 2 } , a r ^ { 4 } , a r ^ { 6 } , \ldots$ is:
(1) $\frac { 25 } { 2 }$
(2) $\frac { 9 } { 2 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 5 } { 2 }$