Question

write a recursive formula and an explicit formula for the follow 6, 14, 22, 30, 38, ... a recursive formula is ( a_1 = 6 ), ( a_n = a_{n - 1} + 8 ) (simplify your answers.) an explicit formula is ( a_n = square ) (simplify your answer.)

Explanation:

Step1: Identify Sequence Type

The sequence \(6, 14, 22, 30, 38, \dots\) is arithmetic. Common difference \(d = 14 - 6 = 8\), first term \(a_1 = 6\).

Step2: Explicit Formula for Arithmetic Sequence

The formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n - 1)d\). Substitute \(a_1 = 6\) and \(d = 8\):
\(a_n = 6 + (n - 1) \times 8\)
Simplify: \(a_n = 6 + 8n - 8 = 8n - 2\).

Answer:

\(8n - 2\)