Question

write a recursive formula and an explicit formula for the following arithmetic sequence
77, 71, 65, 59, 53, ...

a recursive formula is ( a_1 = 77 ), ( a_n = a_{n - 1} + (-6) )
(simplify your answers.)
an explicit formula is ( a_n = 83 - 6n )
(simplify your answer.)

Explanation:

Step1: Analyze the arithmetic sequence

The given arithmetic sequence is \(77, 71, 65, 59, 53, \dots\). The first term \(a_1 = 77\). To find the common difference \(d\), we subtract consecutive terms: \(71 - 77=-6\), \(65 - 71 = -6\), etc. So the common difference \(d=-6\).

Step2: Recursive formula

For an arithmetic sequence, the recursive formula is \(a_1\) (the first term) and \(a_n=a_{n - 1}+d\). Here, \(a_1 = 77\) and \(d=-6\), so \(a_n=a_{n - 1}+(-6)=a_{n - 1}-6\).

Step3: Explicit formula

The explicit formula for an arithmetic sequence is \(a_n=a_1+(n - 1)d\). Substituting \(a_1 = 77\) and \(d=-6\) into the formula:

$$
LATEXBLOCK0
$$

Answer:

The recursive formula is \(a_1 = 77\), \(a_n=a_{n - 1}-6\) (for \(n\geq2\)) and the explicit formula is \(a_n = 83 - 6n\).