Question

1. -2, -4, -8, -16, ...

find $a_9$

1. -3, -9, -27, -81, ...

find $a_{12}$

1. -4, 12, -36, 108, ...

find $a_{11}$

1. -2, -8, -32, -128, ...

find $a_{10}$
given the first term and the common ratio of a geometric sequence find the term named in the problem and the explicit formula.

1. $a_1=-1, r=-5$

find $a_9$

1. $a_1=-1, r=-2$

find $a_{12}$

Explanation:

Geometric Sequence Formula:

The nth term of a geometric sequence is given by:
$$a_n = a_1 \cdot r^{n-1}$$
where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.

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Problem 7:

Step1: Identify $a_1$, $r$, $n$

$a_1=-2$, $r=\frac{-4}{-2}=2$, $n=9$

Step2: Substitute into formula

$a_9 = -2 \cdot 2^{9-1}$

Step3: Simplify the expression

$a_9 = -2 \cdot 2^8 = -2 \cdot 256 = -512$

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Problem 8:

Step1: Identify $a_1$, $r$, $n$

$a_1=-3$, $r=\frac{-9}{-3}=3$, $n=12$

Step2: Substitute into formula

$a_{12} = -3 \cdot 3^{12-1}$

Step3: Simplify the expression

$a_{12} = -3 \cdot 3^{11} = -3^{12} = -531441$

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Problem 9:

Step1: Identify $a_1$, $r$, $n$

$a_1=-4$, $r=\frac{12}{-4}=-3$, $n=11$

Step2: Substitute into formula

$a_{11} = -4 \cdot (-3)^{11-1}$

Step3: Simplify the expression

$a_{11} = -4 \cdot (-3)^{10} = -4 \cdot 59049 = -236196$

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Problem 10:

Step1: Identify $a_1$, $r$, $n$

$a_1=-2$, $r=\frac{-8}{-2}=4$, $n=10$

Step2: Substitute into formula

$a_{10} = -2 \cdot 4^{10-1}$

Step3: Simplify the expression

$a_{10} = -2 \cdot 4^9 = -2 \cdot 262144 = -524288$

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Problem 11:

Step1: Identify $a_1$, $r$, $n$

$a_1=-1$, $r=-5$, $n=9$

Step2: Substitute into formula

$a_9 = -1 \cdot (-5)^{9-1}$

Step3: Simplify the expression

$a_9 = -1 \cdot (-5)^8 = -1 \cdot 390625 = -390625$

Step4: Write explicit formula

$a_n = -1 \cdot (-5)^{n-1} = -(-5)^{n-1}$

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Problem 12:

Step1: Identify $a_1$, $r$, $n$

$a_1=-1$, $r=-2$, $n=12$

Step2: Substitute into formula

$a_{12} = -1 \cdot (-2)^{12-1}$

Step3: Simplify the expression

$a_{12} = -1 \cdot (-2)^{11} = -1 \cdot (-2048) = 2048$

Step4: Write explicit formula

$a_n = -1 \cdot (-2)^{n-1} = -(-2)^{n-1}$

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Answer:

1. $a_9 = -512$
2. $a_{12} = -531441$
3. $a_{11} = -236196$
4. $a_{10} = -524288$
5. $a_9 = -390625$; Explicit formula: $a_n = -(-5)^{n-1}$
6. $a_{12} = 2048$; Explicit formula: $a_n = -(-2)^{n-1}$