Question

determine if the sequence is geometric. if it is, find the explicit formula.

1. $-1, 3, -9, 27, ...$
2. $-4, -24, -144, -864, ...$
3. $2, 6, 18, 54, ...$
4. $1, -5, 25, -125, ...$

Explanation:

For sequence 16) $-1, 3, -9, 27, ...$

Step1: Calculate common ratio $r$

$r = \frac{3}{-1} = -3$, $\frac{-9}{3} = -3$, $\frac{27}{-9} = -3$

Step2: Identify first term $a_1$

$a_1 = -1$

Step3: Write explicit formula

Explicit formula for geometric sequence: $a_n = a_1 r^{n-1}$
$a_n = -1 \times (-3)^{n-1} = (-3)^{n-1} \times (-1) = (-1)^n 3^{n-1}$

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For sequence 17) $-4, -24, -144, -864, ...$

Step1: Calculate common ratio $r$

$r = \frac{-24}{-4} = 6$, $\frac{-144}{-24} = 6$, $\frac{-864}{-144} = 6$

Step2: Identify first term $a_1$

$a_1 = -4$

Step3: Write explicit formula

$a_n = -4 \times 6^{n-1}$

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For sequence 18) $2, 6, 18, 54, ...$

Step1: Calculate common ratio $r$

$r = \frac{6}{2} = 3$, $\frac{18}{6} = 3$, $\frac{54}{18} = 3$

Step2: Identify first term $a_1$

$a_1 = 2$

Step3: Write explicit formula

$a_n = 2 \times 3^{n-1}$

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For sequence 19) $1, -5, 25, -125, ...$

Step1: Calculate common ratio $r$

$r = \frac{-5}{1} = -5$, $\frac{25}{-5} = -5$, $\frac{-125}{25} = -5$

Step2: Identify first term $a_1$

$a_1 = 1$

Step3: Write explicit formula

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

Answer:

1. It is geometric. $a_n = (-1)^n 3^{n-1}$
2. It is geometric. $a_n = -4 \times 6^{n-1}$
3. It is geometric. $a_n = 2 \times 3^{n-1}$
4. It is geometric. $a_n = (-5)^{n-1}$