If $1 + \frac { \sqrt { 3 } - \sqrt { 2 } } { 2 \sqrt { 3 } } + \frac { 5 - 2 \sqrt { 6 } } { 18 } + \frac { 9 \sqrt { 3 } - 11 \sqrt { 2 } } { 36 \sqrt { 3 } } + \frac { 49 - 20 \sqrt { 6 } } { 180 } + \ldots$ upto $\infty = 2 + \left( \sqrt { \frac { b } { a } } + 1 \right) \log _ { e } \left( \frac { a } { b } \right)$, where a and b are integers with $\operatorname { gcd } ( \mathrm { a } , \mathrm { b } ) = 1$, then $11 \mathrm { a } + 18 \mathrm {~b}$ is equal to $\_\_\_\_$

If the range of $f ( \theta ) = \frac { \sin ^ { 4 } \theta + 3 \cos ^ { 2 } \theta } { \sin ^ { 4 } \theta + \cos ^ { 2 } \theta } , \theta \in \mathbb { R }$ is $[ \alpha , \beta ]$, then the sum of the infinite G.P., whose first term is 64 and the common ratio is $\frac { \alpha } { \beta }$, is equal to $\_\_\_\_$

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be a G.P. of increasing positive terms. If $a _ { 1 } a _ { 5 } = 28$ and $a _ { 2 } + a _ { 4 } = 29$, then $a _ { 6 }$ is equal to:
(1) 628
(2) 812
(3) 526
(4) 784

Let $f : (0, \infty) \rightarrow \mathbf{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^{2} f'(x) = 2x f(x) + 3$, with $f(1) = 4$. Then $2f(2)$ is equal to:
(1) 39
(2) 19
(3) 29
(4) 23

Let $S = \mathbf { N } \cup \{ 0 \}$. Define a relation $R$ from $S$ to $\mathbf { R }$ by : $\mathbf { R } = \left\{ ( x , y ) : \log _ { \mathrm { e } } y = x \log _ { \mathrm { e } } \left( \frac { 2 } { 5 } \right) , x \in \mathrm {~S} , y \in \mathbf { R } \right\}$ Then, the sum of all the elements in the range of $R$ is equal to :
(1) $\frac { 10 } { 9 }$
(2) $\frac { 3 } { 2 }$
(3) $\frac { 5 } { 2 }$
(4) $\frac { 5 } { 3 }$

Let $\mathrm { E } _ { 1 } : \frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$ be an ellipse. Ellipses $\mathrm { E } _ { i }$ are constructed such that their centres and eccentricities are same as that of $E _ { 1 }$, and the length of minor axis of $E _ { i }$ is the length of major axis of $E _ { i + 1 } ( i \geq 1 )$. If $A _ { i }$ is the area of the ellipse $E _ { i }$, then $\frac { 5 } { \pi } \left( \sum _ { i = 1 } ^ { \infty } A _ { i } \right)$, is equal to

Q63. In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\frac { 70 } { 3 }$ and the product of the third and fifth terms is 49 . Then the sum of the $4 ^ { \text {th } } , 6 ^ { \text {th } }$ and $8 ^ { \text {th } }$ terms is equal to :
(1) 96
(2) 91
(3) 84
(4) 78

Q63. Let $a , a r , a r ^ { 2 } , \quad$ be an infinite G.P. If $\sum _ { n = 0 } ^ { \infty } a r ^ { n } = 57$ and $\sum _ { n = 0 } ^ { \infty } a ^ { 3 } r ^ { 3 n } = 9747$, then $a + 18 r$ is equal to
(1) 46
(2) 38
(3) 31
(4) 27

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

Q64. Let $A B C$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $A B C$ and the same process is repeated infinitely many times. If P is the sum of perimeters and $Q$ is be the sum of areas of all the triangles formed in this process, then :
(1) $\mathrm { P } ^ { 2 } = 6 \sqrt { 3 } \mathrm { Q }$
(2) $\mathrm { P } ^ { 2 } = 36 \sqrt { 3 } \mathrm { Q }$
(3) $P = 36 \sqrt { 3 } Q ^ { 2 }$
(4) $\mathrm { P } ^ { 2 } = 72 \sqrt { 3 } \mathrm { Q }$

Q87. If the range of $f ( \theta ) = \frac { \sin ^ { 4 } \theta + 3 \cos ^ { 2 } \theta } { \sin ^ { 4 } \theta + \cos ^ { 2 } \theta } , \theta \in \mathbb { R }$ is $[ \alpha , \beta ]$, then the sum of the infinite G.P., whose first term is 64 and the common ratio is $\frac { \alpha } { \beta }$, is equal to $\_\_\_\_$

If $\mathbf{a}_{\mathbf{1}}, \mathbf{a}_{\mathbf{2}}, \mathbf{a}_{\mathbf{3}}, \ldots$ are in increasing geometric progression such that
$a_{1} + a_{3} + a_{5} = 21$,
$a_{1}a_{3}a_{5} = 64$
then $a_{1} + a_{2} + a_{3}$ is
(A) 7 (B) 10 (C) 12 (D) 15

Let product of 3 terms in G.P. is 27. If sum of these 3 terms lies in the interval $\mathbf { R } - \mathbf { ( a , b ) }$, then $\mathbf { a } ^ { \mathbf { 2 } } \boldsymbol { + } \mathbf { b } ^ { \mathbf { 2 } }$ is equal to

In the following, all terms of the sequences under consideration are real numbers.
(1) For a geometric sequence $\left\{ s _ { n } \right\}$ with first term 1 and common ratio 2,
$$s _ { 1 } s _ { 2 } s _ { 3 } = \square , \quad s _ { 1 } + s _ { 2 } + s _ { 3 } = \square$$
(2) Let $\left\{ s _ { n } \right\}$ be a geometric sequence with first term $x$ and common ratio $r$. Let $a, b$ be real numbers (with $a \neq 0$), and suppose the first three terms of $\left\{ s _ { n } \right\}$ satisfy
$$\begin{aligned} & s _ { 1 } s _ { 2 } s _ { 3 } = a ^ { 3 } \\ & s _ { 1 } + s _ { 2 } + s _ { 3 } = b \end{aligned}$$
Then
$$x r = \square$$
Furthermore, using (2) and (3), we find the relation satisfied by $r, a, b$:
$$\text { エ } r ^ { 2 } + ( \text { オ } - \text { カ } ) r + \text { 倍 } = 0$$
Since there exists a real number $r$ satisfying (4),
$$\text { ク } a ^ { 2 } + \text { ケ } a b - b ^ { 2 } \leqq 0$$
Conversely, when $a, b$ satisfy (5), we can find the values of $r, x$ using (3) and (4).
(3) When $a = 64 , b = 336$, consider the geometric sequence $\left\{ s _ { n } \right\}$ satisfying conditions (1) and (2) in (2) with common ratio greater than 1. Using (3) and (4), we find the common ratio $r$ and first term $x$ of $\left\{ s _ { n } \right\}$: $r = \square , x =$ サシ.
Using $\left\{ s _ { n } \right\}$, define the sequence $\left\{ t _ { n } \right\}$ by
$$t _ { n } = s _ { n } \log _ { \square } s _ { n } \quad ( n = 1,2,3 , \cdots )$$
Then the general term of $\left\{ t _ { n } \right\}$ is $t _ { n } = ( n +$ ス $) \cdot$ コ $^ { n + }$ セ. The sum $U _ { n }$ of the first $n$ terms of $\left\{ t _ { n } \right\}$ is found by computing $U _ { n } - \square U _ { n }$:

Given that positive real numbers $a , b , c , d , e$ form a geometric sequence with $a < b < c < d < e$, select the options that form a geometric sequence.
(1) $a , - b , c , - d , e$
(2) $e , d , c , b , a$
(3) $\log a , \log b , \log c , \log d , \log e$
(4) $3 ^ { a } , 3 ^ { b } , 3 ^ { c } , 3 ^ { d } , 3 ^ { e }$
(5) $a b c , b c d , c d e$

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 terms of an infinite series $S$ are formed by adding together the corresponding terms in two infinite geometric series, T and U .
The first term of T and the first term of U are each 4. In order, the first three terms of the combined series $S$ are 8,3 , and $\frac { 5 } { 4 }$ What is the sum to infinity of $S$ ?
A $\frac { 32 } { 5 }$ B $\frac { 20 } { 3 }$ C $\frac { 64 } { 5 }$ D $\frac { 40 } { 3 }$ E 16 F 32

The first term of a geometric progression is $2 \sqrt { 3 }$ and the fourth term is $\frac { 9 } { 4 }$ What is the sum to infinity of this geometric progression?
A $- 2 ( 2 - \sqrt { 3 } )$
B $4 ( 2 \sqrt { 3 } - 3 )$
C $\frac { 16 ( 8 \sqrt { 3 } + 9 ) } { 37 }$
D $\frac { 4 ( 2 \sqrt { 3 } - 3 ) } { 7 }$
$\mathbf { E } \frac { 4 ( 2 \sqrt { 3 } + 3 ) } { 7 }$
F $\quad 2 ( 2 + \sqrt { 3 } )$
G $4 ( 2 \sqrt { 3 } + 3 )$

The sum to infinity of a geometric progression is 6 .
The sum to infinity of the squares of each term in the progression is 12 .
Find the sum to infinity of the cubes of each term in the progression.
A 8
B 18
C 24
D $\quad \frac { 216 } { 7 }$
E 72
F 216

$S$ is a geometric sequence. The sum of the first 6 terms of S is equal to 9 times the sum of the first 3 terms of S. The $7^{\text{th}}$ term of S is 360. Find the $1^{\text{st}}$ term of S.

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

A geometric sequence has first term $a$ and common ratio $r$, where $a$ and $r$ are positive integers and $r$ is greater than 1.
The sum of the first $n$ terms of this sequence is denoted by $S _ { n }$
It is given that the terms of the sequence satisfy
$$S _ { 30 } - S _ { 20 } = k S _ { 10 }$$
for some positive integer $k$.
What is the smallest possible value of $k$ ?

A circle $C _ { n }$ is defined by
$$x ^ { 2 } + y ^ { 2 } = 2 n ( x + y )$$
where $n$ is a positive integer.
$C _ { 1 }$ and $C _ { 2 }$ are drawn and the area between them is shaded.
Next, $C _ { 3 }$ and $C _ { 4 }$ are drawn and the area between them is shaded.
This is shown in the diagram.
This process continues until 100 circles have been drawn.
What is the total shaded area?

You are given that
$$S = 4 + \frac { 8 k } { 7 } + \frac { 16 k ^ { 2 } } { 49 } + \frac { 32 k ^ { 3 } } { 343 } + \cdots + 4 \left( \frac { 2 k } { 7 } \right) ^ { n } + \cdots$$
The value for $k$ is chosen as an integer in the range $- 5 \leq k \leq 5$
All possible values for $k$ are equally likely to be chosen.
What is the probability that the value of $S$ is a finite number greater than 3 ?