Suppose $a, b$ and $c$ are three numbers in G.P. If the equations $ax^2 + 2bx + c = 0$ and $dx^2 + 2ex + f = 0$ have a common root, then $\frac{d}{a}, \frac{e}{b}$ and $\frac{f}{c}$ are in
(A) A.P.
(B) G.P.
(C) H.P.
(D) none of the above.

Suppose $a, b$ and $c$ are three numbers in G.P. If the equations $ax^2 + 2bx + c = 0$ and $dx^2 + 2ex + f = 0$ have a common root, then $\frac{d}{a}, \frac{e}{b}$ and $\frac{f}{c}$ are in
(A) A.P.
(B) G.P.
(C) H.P.
(D) none of the above

Suppose $a, b$ and $c$ are three numbers in G.P. If the equations $a x ^ { 2 } + 2 b x + c = 0$ and $d x ^ { 2 } + 2 e x + f = 0$ have a common root, then $\frac { d } { a } , \frac { e } { b }$ and $\frac { f } { c }$ are in
(A) A.P.
(B) G.P.
(C) H.P.
(D) none of the above

Let $T$ be a right-angled triangle in the plane whose side lengths are in a geometric progression. Let $n(T)$ denote the number of sides of $T$ that have integer lengths. Then the maximum value of $n(T)$ over all such $T$ is
(A) 0
(B) 1
(C) 2
(D) 3

Suppose $a , b$ and $c$ are three numbers in G.P. If the equations $a x ^ { 2 } + 2 b x + c = 0$ and $d x ^ { 2 } + 2 e x + f = 0$ have a common root, then $\frac { d } { a } , \frac { e } { b }$ and $\frac { f } { c }$ are in
(a) A.P.
(B) G.P.
(C) H.P.
(D) none of the above.

15. Let $a$ and $\beta$ be the roots of $x 2 - x + p = 0$ and $y$ and $\delta$ be the roots of $x 2 - 4 x + q = 0$. if $\mathrm { a } , \beta , \mathrm { Y } , \delta$ are in G.P. then the integral values of P and q respectively, are:
(A) $- 2 , - 32$
(B) $- 2,3$
(C) $- 6,3$
(D) $- 6 , - 32$

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 $\_\_\_\_$ .

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) $4 l m n ^ { 2 }$

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:

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) $7 + \sqrt { 3 }$

Let $0 < z < y < x$ be three real numbers such that $\frac { 1 } { x } , \frac { 1 } { y } , \frac { 1 } { z }$ are in an arithmetic progression and $x , \sqrt{2} y , z$ are in a geometric progression. If $x y + y z + z x = \frac { 3 } { \sqrt { 2 } } x y z$, then $3 ( x + y + z ) ^ { 2 }$ is equal to

Let $A _ { 1 }$ and $A _ { 2 }$ be two arithmetic means and $G _ { 1 } , G _ { 2 }$ and $G _ { 3 }$ be three geometric means of two distinct positive numbers. Then $G _ { 1 } ^ { 4 } + G _ { 2 } ^ { 4 } + G _ { 3 } ^ { 4 } + G _ { 1 } ^ { 2 } G _ { 3 } ^ { 2 }$ is equal to
(1) $\left( A _ { 1 } + A _ { 2 } \right) ^ { 2 } G _ { 1 } G _ { 3 }$
(2) $2 \left( A _ { 1 } + A _ { 2 } \right) G _ { 1 } G _ { 3 }$
(3) $\left( A _ { 1 } + A _ { 2 } \right) G _ { 1 } ^ { 2 } G _ { 3 } ^ { 2 }$
(4) $2 \left( A _ { 1 } + A _ { 2 } \right) G _ { 1 } ^ { 2 } G _ { 3 } ^ { 2 }$

For the two positive numbers $a , b$, if $a , b$ and $\frac { 1 } { 18 }$ are in a geometric progression, while $\frac { 1 } { a } , 10$ and $\frac { 1 } { b }$ are in an arithmetic progression, then $16 a + 12 b$ is equal to $\_\_\_\_$.

Let $\alpha$ and $\beta$ be the roots of the equation $px^2 + qx - r = 0$, where $p \neq 0$. If $p, q$ and $r$ be the consecutive terms of a non-constant G.P and $\frac{1}{\alpha} + \frac{1}{\beta} = \frac{3}{4}$, then the value of $\alpha - \beta^2$ is:
(1) $\frac{80}{9}$
(2) 9
(3) $\frac{20}{3}$
(4) 8

Let $2^{\text{nd}}$, $8^{\text{th}}$ and $44^{\text{th}}$ terms of a non-constant A.P. be respectively the $1^{\text{st}}$, $2^{\text{nd}}$ and $3^{\text{rd}}$ terms of G.P. If the first term of A.P. is 1 then the sum of first 20 terms is equal to-
(1) 980
(2) 960
(3) 990
(4) 970

Q64. Let the first three terms $2 , p$ and $q$, with $q \neq 2$, of a G.P. be respectively the $7 ^ { \text {th } } , 8 ^ { \text {th } }$ and $13 ^ { \text {th } }$ terms of an A.P. If the $5 ^ { \text {th } }$ term of the G.P. is the $n ^ { \text {th } }$ term of the A.P., then $n$ is equal to:
(1) 163
(2) 151
(3) 177
(4) 169

19. A geometric series has first term 4 and common ratio $r$, where $0 < r < 1$.
The first, second, and fourth terms of this geometric series form three successive terms of an arithmetic series.
The sum to infinity of the geometric series is
A $\frac { 1 } { 2 } ( \sqrt { 5 } - 1 )$
B $2 ( 3 - \sqrt { 5 } )$
C $2 ( 1 + \sqrt { 5 } )$
D $2 ( 3 + \sqrt { 5 } )$

The $1^{\text{st}}, 2^{\text{nd}}$ and $3^{\text{rd}}$ terms of a geometric progression are also the $1^{\text{st}}, 4^{\text{th}}$ and $6^{\text{th}}$ terms, respectively, of an arithmetic progression.
The sum to infinity of the geometric progression is 12.
Find the $1^{\text{st}}$ term of the geometric progression.
A $1$
B $2$
C $3$
D $4$
E $5$
F $6$

An arithmetic progression and a convergent geometric progression each have first term $\frac { 1 } { 2 }$
The sum of the second terms of the two progressions is $\frac { 1 } { 2 }$ The sum of the third terms of the two progressions is $\frac { 1 } { 8 }$ What is the sum to infinity of the geometric progression?
A - 2 B - 1 C $- \frac { 1 } { 2 }$ D $- \frac { 1 } { 3 }$ E $\frac { 1 } { 3 }$ F $\quad \frac { 1 } { 2 }$ G 1 H 2

The geometric mean of numbers a and b is 3, and their arithmetic mean is 6.
Accordingly, what is the arithmetic mean of $\mathrm { a } ^ { 2 }$ and $\mathrm { b } ^ { 2 }$?
A) 67
B) 65
C) 63
D) 61
E) 57

The geometric mean of real numbers a and b is 4, and the geometric mean of $a - 1$ and $b + 1$ is 6.
Accordingly, what is the difference $\mathbf { a - b }$?
A) 20
B) 21
C) 22
D) 23
E) 24