9. Let $S _ { n }$ denote the sum of the first $n$ terms of the geometric sequence $\left\{ a _ { n } \right\}$. If $S _ { 2 } = 4 , S _ { 4 } = 6$, then $S _ { 6 } =$
A. 7
B. 8
C. 9
D. 10

Given that the sum of the first 3 terms of a geometric sequence $\{a_n\}$ is $168$, and $a_2 - a_5 = 42$, then $a_6 =$
A. $14$
B. $12$
C. $6$
D. $3$

Given that the geometric sequence $\left\{ a _ { n } \right\}$ has the sum of its first 3 terms equal to 168 , and $a _ { 2 } - a _ { 5 } = 42$ , then $a _ { 6 } =$
A. 14
B. 12
C. 6
D. 3

Let $S_n$ denote the sum of the first $n$ terms of the geometric sequence $\{a_n\}$. If $S_4=-5$, $S_6=21S_2$, then $S_8=$
A. 120
B. 85
C. $-85$
D. $-120$

Given that $\left\{ a _ { n } \right\}$ is a geometric sequence with $a _ { 2 } a _ { 4 } a _ { 5 } = a _ { 3 } a _ { 6 }$ and $a _ { 9 } a _ { 10 } = - 8$, then $a _ { 7 } = $ \_\_\_\_

Given that the volumes of three cylinders form a geometric sequence with common ratio 10. The diameter of the first cylinder is 65 mm, the diameters of the second and third cylinders are 325 mm, and the height of the third cylinder is 230 mm. Find the heights of the first two cylinders respectively as \_\_\_\_.

Let $S_n$ denote the sum of the first $n$ terms of a geometric sequence $\{a_n\}$, and let $q$ be the common ratio of $\{a_n\}$, $q > 0$. If $S_3 = 7$, $a_3 = 1$, then
A. $q = \frac{1}{2}$
B. $a_5 = \frac{1}{9}$
C. $S_5 = 8$
D. $a_n + S_n = 8$

If a positive geometric sequence has the sum of its first 4 terms equal to $4$ and the sum of its first 8 terms equal to $68$, then the common ratio of the geometric sequence is $\_\_\_\_$ .

If the sum of the first 4 terms of a geometric sequence is 4 and the sum of the first 8 terms is 68, then the common ratio of the geometric sequence is $\_\_\_\_$ .

Show that a geometric sequence is hypergeometric.

Let $\{C_n\}$ be an infinite sequence of circles lying in the positive quadrant of the $XY$-plane, with strictly decreasing radii and satisfying the following conditions. Each $C_n$ touches both $X$-axis and the $Y$-axis. Further, for all $n \geq 1$, the circle $C_{n+1}$ touches the circle $C_n$ externally. If $C_1$ has radius 10 cm, then show that the sum of the areas of all these circles is $\frac{25\pi}{3\sqrt{2}-4}$ sq. cm.

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

A father wants to distribute a certain sum of money between his daughter and son in such a way that if both of them invest their shares in the scheme that offers compound interest at $\frac { 25 } { 3 } \%$ per annum, for $t$ and $t + 2$ years respectively, then the two shares grow to become equal. If the son's share was rupees 4320, then the total money distributed by the father was
(A) rupees 9360
(B) rupees 9390
(C) rupees 16, 590
(D) rupees 16, 640.

Which of the following is the sum of an infinite geometric sequence whose terms come from the set $\left\{ 1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } , \ldots , \frac { 1 } { 2 ^ { n } } , \ldots \right\}$ ?
(A) $\frac { 1 } { 5 }$
(B) $\frac { 1 } { 7 }$
(C) $\frac { 1 } { 9 }$
(D) $\frac { 1 } { 11 }$

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous function such that $$f ( x + 1 ) = \frac { 1 } { 2 } f ( x ) \text { for all } x \in \mathbb { R } ,$$ and let $a _ { n } = \int _ { 0 } ^ { n } f ( x ) d x$ for all integers $n \geq 1$. Then:
(A) $\lim _ { n \rightarrow \infty } a _ { n }$ exists and equals $\int _ { 0 } ^ { 1 } f ( x ) d x$.
(B) $\lim _ { n \rightarrow \infty } a _ { n }$ does not exist.
(C) $\lim _ { n \rightarrow \infty } a _ { n }$ exists if and only if $\left| \int _ { 0 } ^ { 1 } f ( x ) d x \right| < 1$.
(D) $\lim _ { n \rightarrow \infty } a _ { n }$ exists and equals $2 \int _ { 0 } ^ { 1 } f ( x ) d x$.

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.

Suppose four distinct positive numbers $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 }$ are in G.P. Let $b _ { 1 } = a _ { 1 }$, $b _ { 2 } = b _ { 1 } + a _ { 2 } , b _ { 3 } = b _ { 2 } + a _ { 3 }$ and $b _ { 4 } = b _ { 3 } + a _ { 4 }$. STATEMENT-1 : The numbers $b _ { 1 } , b _ { 2 } , b _ { 3 } , b _ { 4 }$ are neither in A.P. nor in G.P. and
STATEMENT-2 : The numbers $b _ { 1 } , b _ { 2 } , b _ { 3 } , b _ { 4 }$ are in H.P.
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

Let $\mathrm { S } _ { \mathrm { k } } , \mathrm { k } = 1,2 , \ldots , 100$, denote the sum of the infinite geometric series whose first term is $\frac { \mathrm { k } - 1 } { \mathrm { k } ! }$ and the common ratio is $\frac { 1 } { \mathrm { k } }$. Then the value of $\frac { 100 ^ { 2 } } { 100 ! } + \sum _ { \mathrm { k } = 1 } ^ { 100 } \left| \left( \mathrm { k } ^ { 2 } - 3 \mathrm { k } + 1 \right) \mathrm { S } _ { \mathrm { k } } \right|$ is

The number of real solutions of the equation $$\sin ^ { - 1 } \left( \sum _ { i = 1 } ^ { \infty } x ^ { i + 1 } - x \sum _ { i = 1 } ^ { \infty } \left( \frac { x } { 2 } \right) ^ { i } \right) = \frac { \pi } { 2 } - \cos ^ { - 1 } \left( \sum _ { i = 1 } ^ { \infty } \left( - \frac { x } { 2 } \right) ^ { i } - \sum _ { i = 1 } ^ { \infty } ( - x ) ^ { i } \right)$$ lying in the interval $\left( - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$ is $\_\_\_\_$. (Here, the inverse trigonometric functions $\sin ^ { - 1 } x$ and $\cos ^ { - 1 } x$ assume values in $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ and $[ 0 , \pi ]$, respectively.)

For each positive integer $n$, let $$y _ { n } = \frac { 1 } { n } ( ( n + 1 ) ( n + 2 ) \cdots ( n + n ) ) ^ { \frac { 1 } { n } }$$ For $x \in \mathbb { R }$, let $[ x ]$ be the greatest integer less than or equal to $x$. If $\lim _ { n \rightarrow \infty } y _ { n } = L$, then the value of $[ L ]$ is $\_\_\_\_$.

Let
$$\alpha = \sum _ { k = 1 } ^ { \infty } \sin ^ { 2 k } \left( \frac { \pi } { 6 } \right) .$$
Let $g : [ 0,1 ] \rightarrow \mathbb { R }$ be the function defined by
$$g ( x ) = 2 ^ { \alpha x } + 2 ^ { \alpha ( 1 - x ) }$$
Then, which of the following statements is/are TRUE ?
(A) The minimum value of $g ( x )$ is $2 ^ { \frac { 7 } { 6 } }$
(B) The maximum value of $g ( x )$ is $1 + 2 ^ { \frac { 1 } { 3 } }$
(C) The function $g ( x )$ attains its maximum at more than one point
(D) The function $g ( x )$ attains its minimum at more than one point

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 $\_\_\_\_$ .