Question 167
A soma dos termos de uma progressão geométrica finita de razão $q = 2$, primeiro termo $a_1 = 1$ e $n = 5$ termos é
(A) 15 (B) 20 (C) 31 (D) 32 (E) 63

Uma progressão geométrica tem primeiro termo $a_1 = 2$ e razão $q = 3$. A soma dos quatro primeiros termos dessa progressão é
(A) 26 (B) 40 (C) 54 (D) 80 (E) 162

A geometric progression has first term 2 and common ratio 3. What is the sum of the first 4 terms?
(A) 60
(B) 70
(C) 80
(D) 90
(E) 100

For a geometric sequence $\left\{ a _ { n } \right\}$ with common ratio $r$ and $a _ { 2 } = 1$, let $\omega = a _ { 1 } a _ { 2 } a _ { 3 } \cdots a _ { 10 }$ be the product of the first 10 terms. Find the value of $\log _ { r } \omega$. (Here, $r > 0$ and $r \neq 1$.) [3 points]

For a geometric sequence $\left\{ a _ { n } \right\}$, when $a _ { 3 } = 2 , a _ { 6 } = 16$, find the value of $a _ { 9 }$. [3 points]

Four numbers $1 , a , b , c$ form a geometric sequence with common ratio $r$ in this order, and satisfy $\log _ { 8 } c = \log _ { a } b$. What is the value of the common ratio $r$? (where $r > 1$) [3 points]
(1) 2
(2) $\frac { 5 } { 2 }$
(3) 3
(4) $\frac { 7 } { 2 }$
(5) 4

For two natural numbers $a$ and $b$, the three numbers $a ^ { n } , 2 ^ { 4 } \times 3 ^ { 6 } , b ^ { n }$ form a geometric sequence in this order. Find the minimum value of $a b$. (Here, $n$ is a natural number.) [4 points]

For a geometric sequence $\left\{ a _ { n } \right\}$ with all positive terms, $$\frac { a _ { 1 } a _ { 2 } } { a _ { 3 } } = 2 , \quad \frac { 2a _ { 2 } } { a _ { 1 } } + \frac { a _ { 4 } } { a _ { 2 } } = 8$$ what is the value of $a _ { 3 }$? [3 points]
(1) 16
(2) 18
(3) 20
(4) 22
(5) 24

For a geometric sequence $\left\{ a _ { n } \right\}$ with positive common ratio, if $a _ { 1 } = 3 , a _ { 5 } = 48$, what is the value of $a _ { 3 }$? [3 points]
(1) 18
(2) 16
(3) 14
(4) 12
(5) 10

For a geometric sequence $\left\{ a _ { n } \right\}$ with a non-zero first term, $$a _ { 3 } = 4 a _ { 1 } , \quad a _ { 7 } = \left( a _ { 6 } \right) ^ { 2 }$$ what is the value of the first term $a _ { 1 }$? [3 points]
(1) $\frac { 1 } { 16 }$
(2) $\frac { 1 } { 8 }$
(3) $\frac { 3 } { 16 }$
(4) $\frac { 1 } { 4 }$
(5) $\frac { 5 } { 16 }$

When three numbers $\frac { 9 } { 4 } , a , 4$ form a geometric sequence in this order, what is the value of the positive number $a$? [3 points]
(1) $\frac { 8 } { 3 }$
(2) 3
(3) $\frac { 10 } { 3 }$
(4) $\frac { 11 } { 3 }$
(5) 4

For a geometric sequence $\left\{ a _ { n } \right\}$ with first term 7, let $S _ { n }$ denote the sum of the first $n$ terms. $$\frac { S _ { 9 } - S _ { 5 } } { S _ { 6 } - S _ { 2 } } = 3$$ Find the value of $a _ { 7 }$. [3 points]

For a geometric sequence $\left\{ a _ { n } \right\}$ with all positive terms, $$\frac { a _ { 16 } } { a _ { 14 } } + \frac { a _ { 8 } } { a _ { 7 } } = 12$$ Find the value of $\frac { a _ { 3 } } { a _ { 1 } } + \frac { a _ { 6 } } { a _ { 3 } }$. [3 points]

For a geometric sequence $\left\{ a _ { n } \right\}$ with first term $\frac { 1 } { 8 }$, if $\frac { a _ { 3 } } { a _ { 2 } } = 2$, what is the value of $a _ { 5 }$? [2 points]
(1) $\frac { 1 } { 4 }$
(2) $\frac { 1 } { 2 }$
(3) 1
(4) 2
(5) 4

A sequence $\left\{ a _ { n } \right\}$ satisfies the following conditions.
(a) $\left| a _ { 1 } \right| = 2$
(b) For all natural numbers $n$, $\left| a _ { n + 1 } \right| = 2 \left| a _ { n } \right|$.
(c) $\sum _ { n = 1 } ^ { 10 } a _ { n } = - 14$ Find the value of $a _ { 1 } + a _ { 3 } + a _ { 5 } + a _ { 7 } + a _ { 9 }$. [4 points]

A geometric sequence $\left\{ a _ { n } \right\}$ with positive common ratio satisfies $$a _ { 2 } + a _ { 4 } = 30 , \quad a _ { 4 } + a _ { 6 } = \frac { 15 } { 2 }$$ What is the value of $a _ { 1 }$? [3 points]
(1) 48
(2) 56
(3) 64
(4) 72
(5) 80

Let $S_n$ denote the sum of the first $n$ terms of a geometric sequence $\{a_n\}$. $$S_4 - S_2 = 3a_4, \quad a_5 = \frac{3}{4}$$ Find the value of $a_1 + a_2$. [3 points]
(1) 27
(2) 24
(3) 21
(4) 18
(5) 15

A geometric sequence $\left\{ a_{n} \right\}$ with first term and common ratio both equal to a positive number $k$ satisfies $$\frac{a_{4}}{a_{2}} + \frac{a_{2}}{a_{1}} = 30$$ What is the value of $k$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

A geometric sequence $\left\{ a _ { n } \right\}$ satisfies $$2 \left( a _ { 1 } + a _ { 4 } + a _ { 7 } \right) = a _ { 4 } + a _ { 7 } + a _ { 10 } = 6$$ What is the value of $a _ { 10 }$? [4 points]
(1) $\frac { 22 } { 7 }$
(2) $\frac { 24 } { 7 }$
(3) $\frac { 26 } { 7 }$
(4) $\frac { 30 } { 7 }$
(5) $\frac { 32 } { 7 }$

18. The sequence $\left\{ a _ { n } \right\}$ is an increasing geometric sequence with $a _ { 1 } + a _ { 4 } = 9 , a _ { 2 } a _ { 3 } = 8$.
(1) Find the general term formula of the sequence $\left\{ a _ { n } \right\}$;
(2) Let $S _ { n }$ be the sum of the first $n$ terms of the sequence $\left\{ a _ { n } \right\}$, and $b _ { n } = \frac { a _ { n + 1 } } { S _ { n } S _ { n + 1 } }$. Find the sum $T _ { n }$ of the first $n$ terms of the sequence $\left\{ b _ { n } \right\}$.

9. Given a geometric sequence $\left\{ a _ { n } \right\}$ satisfying $a _ { 1 } = \frac { 1 } { 4 } , a _ { 3 } a _ { 5 } = 4 \left( a _ { 4 } - 1 \right)$, then $a _ { 2 } =$
A. $2$
B. $1$
C. $\frac { 1 } { 2 }$
D. $\frac { 1 } { 8 }$

A geometric sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 3 , a _ { 1 } + a _ { 3 } + a _ { 5 } = 21$ , then $a _ { 3 } + a _ { 5 } + a _ { 7 } =$
(A) $21$
(B) $42$
(C) $63$
(D) $84$

Let $S _ { n }$ denote the sum of the first $n$ terms of the sequence $\left\{ a _ { n } \right\}$. If $S _ { n } = 2 a _ { n } + 1$, then $S _ { 6 } = \_\_\_\_$

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

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 $S _ { 3 } = \_\_\_\_$.