Question

write a recursive formula for each sequence.
57, 44, 31, 18, ...
$a_1 = \square$
$a_n = \square$ for $n \geq 2$

10, 1, $\frac{1}{10}$, $\frac{1}{100}$, ...
$a_1 = \square$
$a_n = \square$ for $n \geq 2$

Explanation:

Step1: Identify first term (left seq)

$a_1 = 57$

Step2: Find common difference (left seq)

$44-57=-13$, $31-44=-13$, so $a_n = a_{n-1} - 13$ for $n\geq2$

Step3: Identify first term (right seq)

$a_1 = 10$

Step4: Find common ratio (right seq)

$\frac{1}{10}=1\times\frac{1}{10}$, $\frac{1}{100}=\frac{1}{10}\times\frac{1}{10}$, so $a_n = a_{n-1} \times \frac{1}{10}$ for $n\geq2$

Answer:

For the sequence 57, 44, 31, 18, ...:
$a_1 = 57$
$a_n = a_{n-1} - 13$ for $n\geq2$

For the sequence $10, 1, \frac{1}{10}, \frac{1}{100}, ...$:
$a_1 = 10$
$a_n = \frac{a_{n-1}}{10}$ for $n\geq2$