A soma infinita da progressão geométrica com primeiro termo $a_1 = 6$ e razão $q = \dfrac{1}{3}$ é
(A) 6 (B) 7 (C) 8 (D) 9 (E) 10

For two infinite geometric sequences $\left\{ a _ { n } \right\} , \left\{ b _ { n } \right\}$ with the same common ratio, $a _ { 1 } - b _ { 1 } = 1$, $\sum _ { n = 1 } ^ { \infty } a _ { n } = 8$, and $\sum _ { n = 1 } ^ { \infty } b _ { n } = 6$. Find the value of $\sum _ { n = 1 } ^ { \infty } a _ { n } b _ { n }$. [3 points]

A geometric sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 2 } = \frac { 1 } { 2 }$ and $a _ { 5 } = \frac { 1 } { 6 }$. When $\sum _ { n = 1 } ^ { \infty } a _ { n } a _ { n + 1 } a _ { n + 2 } = \frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

On the coordinate plane, for a natural number $n$, the coordinates of point $\mathrm { P } _ { n }$ are $\left( n , 3 ^ { n } \right)$ and the coordinates of point $\mathrm { Q } _ { n }$ are $( n , 0 )$. Let $a _ { n }$ be the area of the quadrilateral $\mathrm { P } _ { n } \mathrm { Q } _ { n + 1 } \mathrm { Q } _ { n + 2 } \mathrm { P } _ { n + 1 }$. When $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { a _ { n } } = \frac { q } { p }$, find the value of $p ^ { 2 } + q ^ { 2 }$. (where $p$ and $q$ are coprime natural numbers) [4 points]

For a geometric sequence $\left\{ a _ { n } \right\}$ with $a _ { 1 } = 3 , a _ { 2 } = 1$, what is the value of $\sum _ { n = 1 } ^ { \infty } \left( a _ { n } \right) ^ { 2 }$? [3 points]
(1) $\frac { 81 } { 8 }$
(2) $\frac { 83 } { 8 }$
(3) $\frac { 85 } { 8 }$
(4) $\frac { 87 } { 8 }$
(5) $\frac { 89 } { 8 }$

For two sequences $\left\{ a _ { n } \right\}$ and $\left\{ b _ { n } \right\}$, $$\sum _ { n = 1 } ^ { \infty } a _ { n } = 4 , \quad \sum _ { n = 1 } ^ { \infty } b _ { n } = 10$$ find the value of $\sum _ { n = 1 } ^ { \infty } \left( a _ { n } + 5 b _ { n } \right)$. [3 points]

For a geometric sequence $\left\{ a _ { n } \right\}$ with $a _ { 1 } = 3 , a _ { 2 } = 1$, what is the value of $\sum _ { n = 1 } ^ { \infty } \left( a _ { n } \right) ^ { 2 }$? [3 points]
(1) $\frac { 89 } { 8 }$
(2) $\frac { 87 } { 8 }$
(3) $\frac { 85 } { 8 }$
(4) $\frac { 83 } { 8 }$
(5) $\frac { 81 } { 8 }$

A geometric sequence $\left\{ a_{n} \right\}$ satisfies $$\sum_{n=1}^{\infty} \left(\left| a_{n} \right| + a_{n}\right) = \frac{40}{3}, \quad \sum_{n=1}^{\infty} \left(\left| a_{n} \right| - a_{n}\right) = \frac{20}{3}$$ The inequality $$\lim_{n \rightarrow \infty} \sum_{k=1}^{2n} \left((-1)^{\frac{k(k+1)}{2}} \times a_{m+k}\right) > \frac{1}{700}$$ is satisfied. What is the sum of all natural numbers $m$ satisfying this inequality? [4 points]

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 $\mathrm { S } _ { \mathrm { k } } , \mathrm { k } = 1,2 , \ldots , 100$, 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

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) $( 0,10 )$

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) $( - 10,0 )$

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 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) $\frac { 5 } { 2 }$

Let $S = 2 + \frac { 6 } { 7 } + \frac { 12 } { 7 ^ { 2 } } + \frac { 20 } { 7 ^ { 3 } } + \frac { 30 } { 7 ^ { 4 } } + \ldots$. then $4 S$ is equal to
(1) $\left( \frac { 7 } { 2 } \right) ^ { 2 }$
(2) $\left( \frac { 7 } { 3 } \right) ^ { 3 }$
(3) $\frac { 7 } { 3 }$
(4) $\left( \frac { 7 } { 3 } \right) ^ { 4 }$

Let $\left\{ a _ { k } \right\}$ and $\left\{ b _ { k } \right\} , k \in \mathbb { N }$, be two G.P.s with common ratio $r _ { 1 }$ and $r _ { 2 }$ respectively such that $\mathrm { a } _ { 1 } = \mathrm { b } _ { 1 } = 4$ and $\mathrm { r } _ { 1 } < \mathrm { r } _ { 2 }$. Let $\mathrm { c } _ { \mathrm { k } } = \mathrm { a } _ { \mathrm { k } } + \mathrm { b } _ { \mathrm { k } } , \mathrm { k } \in \mathbb { N }$. If $\mathrm { c } _ { 2 } = 5$ and $\mathrm { c } _ { 3 } = \frac { 13 } { 4 }$ then $\sum _ { \mathrm { k } = 1 } ^ { \infty } \mathrm { c } _ { \mathrm { k } } - \left( 12 \mathrm { a } _ { 6 } + 8 \mathrm {~b} _ { 4 } \right)$ 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 $a , a r , a r ^ { 2 } , \quad$ 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 $7 = 5 + \frac{1}{7}(5 + \alpha) + \frac{1}{7^{2}}(5 + 2\alpha) + \frac{1}{7^{3}}(5 + 3\alpha) + \cdots \infty$, then the value of $\alpha$ is:
(1) $\frac{6}{7}$
(2) 6
(3) $\frac{1}{7}$
(4) 1

Let $\mathrm { E } _ { 1 } : \frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$ 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 $S = \mathbf { N } \cup \{ 0 \}$. 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) $\frac { 5 } { 3 }$