Question

write a recursive formula for each sequence.
2, -6, 18, -54, ...
$a_1 = \square$
$a_n = \square$ for $n \geq 2$
6, 10, 14, 18, ...
$a_1 = \square$
$a_n = \square$ for $n \geq 2$

Explanation:

Sequence 1: 2, -6, 18, -54, ...

Step1: Identify first term

$a_1 = 2$

Step2: Find common ratio

$\frac{-6}{2} = -3$, $\frac{18}{-6} = -3$, $\frac{-54}{18} = -3$
Recursive formula: $a_n = a_{n-1} \times (-3)$ for $n \geq 2$

Sequence 2: 6, 10, 14, 18, ...

Step1: Identify first term

$a_1 = 6$

Step2: Find common difference

$10-6=4$, $14-10=4$, $18-14=4$
Recursive formula: $a_n = a_{n-1} + 4$ for $n \geq 2$

Answer:

For sequence 2, -6, 18, -54, ...:
$a_1 = 2$
$a_n = -3a_{n-1}$ for $n \geq 2$

For sequence 6, 10, 14, 18, ...:
$a_1 = 6$
$a_n = a_{n-1} + 4$ for $n \geq 2$