14. Let $S _ { n }$ denote the sum of the first $n$ terms of a geometric sequence $\left\{ a _ { n } \right\}$ . If $a _ { 1 } = \frac { 1 } { 3 } , a _ { 4 } ^ { 2 } = a _ { 6 }$ , then $S _ { 5 } =$ $\_\_\_\_$ .

6. A geometric sequence $\left\{ a _ { n } \right\}$ with all positive terms has the sum of its first 4 terms equal to 15, and $a _ { 5 } = 3 a _ { 3 } + 4 a _ { 1 }$ . Then $a _ { 3 } =$
A. 16
B. 8
C. 4
D. 2

5. A geometric sequence $\left\{ a _ { n } \right\}$ with all positive terms has the sum of its first 4 terms equal to 15, and $a _ { 5 } = 3 a _ { 3 } + 4 a _ { 1 }$ . Then $a _ { 3 } =$
A. 16
B. 8
C. 4
D. 2

Let $\left\{ a _ { n } \right\}$ be a geometric sequence with $a _ { 1 } + a _ { 2 } + a _ { 3 } = 1 , a _ { 2 } + a _ { 3 } + a _ { 4 } = 2$ , then $a _ { 6 } + a _ { 7 } + a _ { 8 } =$
A. 12
B. 24
C. 30
D. 32

In the sequence $\left\{ a _ { n } \right\}$ , $a _ { 1 } = 2 , a _ { m + n } = a _ { m } a _ { n }$ . If $a _ { k + 1 } + a _ { k + 2 } + \cdots + a _ { k + 10 } = 2 ^ { 15 } - 2 ^ { 5 }$ , then $k =$
A． 2
B． 3
C． 4
D． 5

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 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

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

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 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 \_\_\_\_.

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 $\_\_\_\_$ .

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals
(1) $\frac { 1 } { 2 } ( 1 - \sqrt { 5 } )$
(2) $\frac { 1 } { 2 } \sqrt { 5 }$
(3) $\sqrt { 5 }$
(4) $\frac { 1 } { 2 } ( \sqrt { 5 } - 1 )$

The difference between the fourth term and the first term of a Geometrical Progression is 52. If the sum of its first three terms is 26, then the sum of the first six terms of the progression is
(1) 63
(2) 189
(3) 728
(4) 364

The sum of first 20 terms of the sequence $0.7, 0.77, 0.777, \ldots$ is
(1) $\frac{7}{81}(179-10^{-20})$
(2) $\frac{7}{9}(99-10^{-20})$
(3) $\frac{7}{81}(179+10^{-20})$
(4) $\frac{7}{9}(99+10^{-20})$

The sum of the first 20 terms of the series $1 + \frac { 3 } { 2 } + \frac { 7 } { 4 } + \frac { 15 } { 8 } + \frac { 31 } { 16 } + \ldots$ is
(1) $39 + \frac { 1 } { 2 ^ { 19 } }$
(2) $38 + \frac { 1 } { 2 ^ { 20 } }$
(3) $38 + \frac { 1 } { 2 ^ { 19 } }$
(4) $39 + \frac { 1 } { 2 ^ { 20 } }$

If $\alpha , \beta$ and $\gamma$ are three consecutive terms of a non-constant G.P. Such that the equations $\alpha x ^ { 2 } + 2 \beta x + \gamma = 0$ and $x ^ { 2 } + x - 1 = 0$ have a common root, then $\alpha ( \beta + \gamma )$ is equal to:
(1) $\beta \gamma$
(2) $\alpha \beta$
(3) $\alpha \gamma$
(4) 0

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$, be a G.P. such that $a _ { 1 } < 0 , a _ { 1 } + a _ { 2 } = 4$ and $a _ { 3 } + a _ { 4 } = 16$. If $\sum _ { i = 1 } ^ { 9 } a _ { i } = 4 \lambda$, then $\lambda$, is equal to.
(1) - 513
(2) - 171
(3) 171
(4) $\frac { 511 } { 3 }$

Let $a _ { n }$ be the $n ^ { \text {th } }$ term of a G.P. of positive terms. If $\sum _ { n = 1 } ^ { 100 } a _ { 2 n + 1 } = 200$ and $\sum _ { n = 1 } ^ { 100 } a _ { 2 n } = 100$, then $\sum _ { n = 1 } ^ { 200 } a _ { n }$ is equal to:
(1) 300
(2) 225
(3) 175
(4) 150

The sum of the first three terms of G.P is $S$ and their products is 27. Then all such $S$ lie in
(1) $(-\infty, -9] \cup [3, \infty)$
(2) $[-3, \infty)$
(3) $(-\infty, -3] \cup [9, \infty)$
(4) $(-\infty, 9]$

Let $S$ be the sum of the first 9 term of the series : $\{ x + k a \} + \left\{ x ^ { 2 } + ( k + 2 ) a \right\} + \left\{ x ^ { 3 } + ( k + 4 ) a \right\} + \left\{ x ^ { 4 } + ( k + 6 ) a \right\} + \ldots$ where $a \neq 0$ and $x \neq 1$. If $S = \frac { x ^ { 10 } - x + 45 a ( x - 1 ) } { x - 1 }$, then $k$ is equal to
(1) - 5
(2) 1
(3) - 3
(4) 3

If $2 ^ { 10 } + 2 ^ { 9 } \cdot 3 ^ { 1 } + 2 ^ { 8 } \cdot 3 ^ { 2 } + \ldots\ldots + 2 \cdot 3 ^ { 9 } + 3 ^ { 10 } = S - 2 ^ { 11 }$, then $S$ is equal to
(1) $3 ^ { 11 } - 2 ^ { 12 }$
(2) $3 ^ { 11 }$
(3) $\frac { 3 ^ { 11 } } { 2 } + 2 ^ { 10 }$
(4) $2.3 ^ { 11 }$

If the sum of the second, third and fourth terms of a positive term G.P. is 3 and the sum of its sixth, seventh and eighth terms is 243, then the sum of the first 50 terms of this G.P. is:
(1) $\frac{1}{26}\left(3^{49}-1\right)$
(2) $\frac{1}{26}\left(3^{50}-1\right)$
(3) $\frac{2}{13}\left(3^{50}-1\right)$
(4) $\frac{1}{13}\left(3^{50}-1\right)$

If $f ( x + y ) = f ( x ) f ( y )$ and $\sum_{x=1}^{n} f(x) = 2$, then the value of $\sum_{x=1}^{n} f(x)$ is given. [Content truncated in source]