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

If each term of a geometric progression $\mathrm { a } _ { 1 } , \mathrm { a } _ { 2 } , \mathrm { a } _ { 3 } , \ldots$ with $\mathrm { a } _ { 1 } = \frac { 1 } { 8 }$ and $\mathrm { a } _ { 2 } \neq \mathrm { a } _ { 1 }$, is the arithmetic mean of the next two terms and $\mathrm { S } _ { \mathrm { n } } = \mathrm { a } _ { 1 } + \mathrm { a } _ { 2 } + \ldots + \mathrm { a } _ { \mathrm { n } }$, then $\mathrm { S } _ { 20 } - \mathrm { S } _ { 18 }$ is equal to
(1) $2 ^ { 15 }$
(2) $- 2 ^ { 18 }$
(3) $2 ^ { 18 }$
(4) $- 2 ^ { 15 }$

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

Let $ABC$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $ABC$ and the same process is repeated infinitely many times. If P is the sum of perimeters and $Q$ is 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) $\mathrm { P } ^ { 2 } = 72 \sqrt { 3 } \mathrm { Q }$

If three successive terms of a G.P. with common ratio $r$ ($r > 1$) are the lengths of the sides of a triangle and $[r]$ denotes the greatest integer less than or equal to $r$, then $3[r] + [-r]$ 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

If $7 = 5 + \frac{1}{7}(5 + \alpha) + \frac{1}{7^{2}}(5 + 2\alpha) + \frac{1}{7^{3}}(5 + 3\alpha) + \cdots \infty$, then the value of $\alpha$ is:
(1) $\frac{6}{7}$
(2) 6
(3) $\frac{1}{7}$
(4) 1

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

Let the sequence $\left\{ a _ { n } \right\} ( n = 1,2,3 , \cdots )$ be an arithmetic progression satisfying
$$a _ { 2 } = 2 , \quad a _ { 6 } = 3 a _ { 3 } .$$
Then, consider the series $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } } { r ^ { a _ { n } } }$, where $r$ is a positive real number.
(1) When we denote the first term of $\left\{ a _ { n } \right\}$ by $a$, and the common difference by $d$, we have
$$a = \mathbf { A B } , \quad d = \mathbf { C } .$$
(2) The series $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } } { r ^ { a _ { n } } }$ is an infinite geometric series where the first term is $\square r^{\mathbf{E}}$, and the common ratio is $\dfrac { \mathbf { F } } { r^{\mathbf{G}} }$. Hence, this series converges when
$$r > 3 ^ { \frac { \mathbf { H } } { \mathbf{I} } } ,$$
and its sum $S$ is
$$S = \frac { \square \mathbf { J } \, r ^ { \mathbf { K } } } { r ^ { \mathbf { L } } - \mathbf { M } } .$$
(3) This sum $S$ is minimized at
$$r = \mathbf { N } ^ { \frac { \mathbf { O } } { 2 } } .$$

A sequence $a _ { 1 } , a _ { 2 } , \cdots$ where the odd-indexed terms form a geometric sequence with common ratio $\frac { 1 } { 3 }$ , and the even-indexed terms form a geometric sequence with common ratio $\frac { 1 } { 2 }$ , with $a _ { 1 } = 3 , a _ { 2 } = 2$ . Select the correct options.
(1) $a _ { 4 } > a _ { 5 } > a _ { 6 } > a _ { 7 }$
(2) $\frac { a _ { 10 } } { a _ { 11 } } > 10$
(3) $\lim _ { n \rightarrow \infty } a _ { n } = 0$
(4) $\lim _ { n \rightarrow \infty } \frac { a _ { n + 1 } } { a _ { n } } = 0$
(5) $\sum _ { n = 1 } ^ { 100 } a _ { n } > 9$

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) $a b c , b c d , c d e$

$$\sum_{n=0}^{100} 3^{n}$$
What is the remainder when this sum is divided by 5?
A) 0
B) 1
C) 2
D) 3
E) 4

The angle formed by the lines $d_{1}$ and $d_{2}$ given above measures $30^{\circ}$. First, a perpendicular $A_{1}B_{1}$ is drawn from point $A_{1}$ on line $d_{1}$ to line $d_{2}$. Then, a perpendicular $B_{1}A_{2}$ is drawn from point $B_{1}$ to line $d_{1}$, and a perpendicular $A_{2}B_{2}$ is drawn from the foot of the perpendicular $A_{2}$ to line $d_{2}$, and this process continues.
Given that $|A_{1}B_{1}| = 12$ cm, what is the sum of the lengths of all perpendiculars drawn to line $d_{2}$ in this manner, $|A_{1}B_{1}| + |A_{2}B_{2}| + |A_{3}B_{3}| + \cdots$, in cm?
A) 32
B) 36
C) 38
D) 40
E) 48

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

An equilateral triangle ABC with side length 1 unit has points D and E marked on sides AB and AC respectively, where these sides are divided into three equal parts. Let K be the midpoint of the line segment DE. A new equilateral triangle is drawn with one vertex at K and the opposite side on BC, and the same process is applied to the newly drawn equilateral triangles.
What is the sum of the areas of all nested triangular regions drawn in this manner, in square units?
A) $\frac { \sqrt { 3 } } { 3 }$
B) $\frac { 3 \sqrt { 3 } } { 4 }$
C) $\frac { 8 \sqrt { 3 } } { 9 }$
D) $\frac { 5 \sqrt { 3 } } { 16 }$
E) $\frac { 9 \sqrt { 3 } } { 32 }$

Below, a sequence of circles drawn side by side is given. In this sequence; the radius of the first circle is 4 units and the radius of each subsequent circle is half the radius of the previous circle.
What is the sum of the circumferences of all circles in this sequence in units?
A) $15 \pi$
B) $16 \pi$
C) $18 \pi$
D) $\frac { 31 \pi } { 2 }$
E) $\frac { 33 \pi } { 2 }$

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

The first three terms of a geometric sequence are $\mathbf { a } + \mathbf { 3 }$, a, and $\mathbf { a } - \mathbf { 2 }$ respectively. Accordingly, what is the fourth term?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 3 }$
C) $\frac { 8 } { 3 }$
D) $\frac { 9 } { 4 }$
E) $\frac { 11 } { 6 }$

An equilateral triangle is inscribed in circle $\mathrm { C } _ { 1 }$ with radius 1 unit as shown in the figure. Let $\mathrm { C } _ { 2 }$ be the circle passing through the midpoints of the sides of this triangle. The same operation is performed for circle $\mathrm { C } _ { 2 }$ and circle $\mathrm { C } _ { 3 }$ is obtained. By repeating this process infinitely many times, the sequence of circles $C _ { 1 } , C _ { 2 } , C _ { 3 } , \cdots$ is obtained.
For each positive integer $\mathbf { k }$, let $\mathbf { A } _ { \mathbf { k } }$ be the area bounded by circle $\mathbf { C } _ { \mathbf { k } }$ in square units. Accordingly, what is the result of the sum
$$\sum _ { k = 1 } ^ { \infty } A _ { k }$$
?
A) $\frac { 3 \pi } { 2 }$
B) $\frac { 4 \pi } { 3 }$
C) $\frac { 5 \pi } { 4 }$
D) $\frac { 6 \pi } { 5 }$
E) $\frac { 9 \pi } { 8 }$

Let x be a positive integer such that
$$\frac { 10 x } { x + 3 }$$
is equal to the square of an integer. What is the sum of the values that x can take?
A) 26
B) 27
C) 29
D) 31
E) 32

In the rectangular coordinate plane, isosceles triangles are drawn with base vertices at consecutive even natural numbers on the x-axis and apex on the curve $y = 2 ^ { - x }$.
Accordingly, what is the sum of the areas of all the triangles drawn in square units?
A) $\frac { 3 } { 2 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 4 } { 3 }$
D) 1
E) 2

Let $R$ be the set of real numbers. For every natural number n,
$$\begin{aligned} & f _ { n } : [ n \pi , ( n + 1 ) \pi ] \rightarrow R \\ & f _ { n } ( x ) = \frac { 1 } { 5 ^ { n } } | \sin x | \end{aligned}$$
What is the sum of the areas of the regions between the functions defined in this form and the x-axis in square units?
A) $\frac { 7 } { 5 }$
B) $\frac { 8 } { 5 }$
C) $\frac { 9 } { 5 }$
D) $\frac { 3 } { 2 }$
E) $\frac { 5 } { 2 }$

Let $(a_n)$ be a geometric sequence. The equality
$$\frac { a _ { 5 } - a _ { 1 } } { \left( a _ { 3 } \right) ^ { 2 } - \left( a _ { 1 } \right) ^ { 2 } } = \frac { 4 } { 9 }$$
is given. Given that $a _ { 2 } = \frac { 3 } { 2 }$, what is $a _ { 4 }$?
A) $\frac { 2 } { 3 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 1 } { 6 }$
D) $\frac { 27 } { 8 }$
E) $\frac { 27 } { 4 }$