Question

ch 6a geometric sequences 2
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determine if the sequence is geometric. if it is, find the common ratio and the three terms in the sequence after the last one given.

1. $-1, -2, -4, -8, \dots$
2. $2, 4, 8, 16, \dots$
3. $4, -16, 64, -256, \dots$
4. $4, 16, 36, 64, \dots$

determine if the sequence is geometric. if it is, find the common ratio, the term named in the problem, and the general formula ($a_n$).

1. $-3, 6, -12, 24, \dots$

find $a_9$

1. $2, 4, 7, 11, \dots$

find $a_{11}$

Explanation:

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For Problem 1: $-1, -2, -4, -8, ...$

Step1: Check if geometric

Calculate ratios: $\frac{-2}{-1}=2$, $\frac{-4}{-2}=2$, $\frac{-8}{-4}=2$

Step2: Identify common ratio

Common ratio $r=2$

Step3: Find next 3 terms

Multiply last term by $r$:
$-8 \times 2 = -16$
$-16 \times 2 = -32$
$-32 \times 2 = -64$

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For Problem 2: $2, 4, 8, 16, ...$

Step1: Check if geometric

Calculate ratios: $\frac{4}{2}=2$, $\frac{8}{4}=2$, $\frac{16}{8}=2$

Step2: Identify common ratio

Common ratio $r=2$

Step3: Find next 3 terms

Multiply last term by $r$:
$16 \times 2 = 32$
$32 \times 2 = 64$
$64 \times 2 = 128$

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For Problem 3: $4, -16, 64, -256, ...$

Step1: Check if geometric

Calculate ratios: $\frac{-16}{4}=-4$, $\frac{64}{-16}=-4$, $\frac{-256}{64}=-4$

Step2: Identify common ratio

Common ratio $r=-4$

Step3: Find next 3 terms

Multiply last term by $r$:
$-256 \times (-4) = 1024$
$1024 \times (-4) = -4096$
$-4096 \times (-4) = 16384$

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For Problem 4: $4, 16, 36, 64, ...$

Step1: Check if geometric

Calculate ratios: $\frac{16}{4}=4$, $\frac{36}{16}=2.25$, $\frac{64}{36}\approx1.78$
Ratios are not equal, so not geometric.

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For Problem 5: $-3, 6, -12, 24, ...$; Find $a_{10}$

Step1: Check if geometric

Calculate ratios: $\frac{6}{-3}=-2$, $\frac{-12}{6}=-2$, $\frac{24}{-12}=-2$

Step2: Identify common ratio

Common ratio $r=-2$

Step3: General formula for $a_n$

Geometric sequence formula: $a_n = a_1 r^{n-1}$
Substitute $a_1=-3, r=-2$:
$a_n = -3(-2)^{n-1}$

Step4: Calculate $a_{10}$

Substitute $n=10$:
$a_{10} = -3(-2)^{10-1} = -3(-2)^9 = -3(-512) = 1536$

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For Problem 6: $2, 4, 7, 11, ...$; Find $a_{11}$

Step1: Check if geometric

Calculate ratios: $\frac{4}{2}=2$, $\frac{7}{4}=1.75$, $\frac{11}{7}\approx1.57$
Ratios are not equal, so not geometric. (This is an arithmetic sequence with increasing differences, so no common ratio, general geometric formula, or valid $a_{11}$ via geometric rules.)

Answer:

1. It is geometric. Common ratio $r=2$. Next three terms: $-16, -32, -64$
2. It is geometric. Common ratio $r=2$. Next three terms: $32, 64, 128$
3. It is geometric. Common ratio $r=-4$. Next three terms: $1024, -4096, 16384$
4. This sequence is not geometric.
5. It is geometric. Common ratio $r=-2$. General formula: $a_n = -3(-2)^{n-1}$. $a_{10}=1536$
6. This sequence is not geometric.