Question

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

a recursive formula is $a_1 = \boxed{77}$, $a_n = \square$
(simplify your answers.)

an explicit formula is $a_n = \square$
(simplify your answer.)

Explanation:

Step1: Identify the first term and common difference

For an arithmetic sequence, the first term \(a_1\) is the first number in the sequence. Here, \(a_1 = 77\). The common difference \(d\) is found by subtracting consecutive terms. Let's take \(71 - 77=-6\), \(65 - 71 = -6\), \(59 - 65=-6\), \(53 - 59=-6\). So \(d=-6\).

Step2: Write the recursive formula

The recursive formula for an arithmetic sequence is \(a_1\) (the first term) and \(a_n=a_{n - 1}+d\) for \(n\gt1\). We know \(a_1 = 77\) and \(d=-6\), so the recursive formula is \(a_1 = 77\), \(a_n=a_{n - 1}-6\) (since \(d = - 6\)).

Step3: Write the 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, we get \(a_n=77+(n - 1)(-6)\). Simplify this: \(a_n=77-6n + 6=83-6n\).

Answer:

Recursive formula: \(a_1 = 77\), \(a_n=a_{n - 1}-6\) (for \(n\gt1\))
Explicit formula: \(a_n = 83-6n\)