Question

write a recursive formula and an explicit formula for the following arithmetic sequence.
6, 14, 22, 30, 38, ...
a recursive formula is ( a_1 = square ), ( a_n = square ).
(simplify your answers.)

Explanation:

Step1: Identify the first term and common difference

The first term \(a_1\) of the arithmetic sequence \(6, 14, 22, 30, 38, \dots\) is \(6\). To find the common difference \(d\), subtract consecutive terms: \(14 - 6 = 8\), \(22 - 14 = 8\), etc. So \(d = 8\).

Step2: Write the recursive formula

For an arithmetic sequence, the recursive formula is \(a_1=\text{first term}\) and \(a_n = a_{n - 1}+d\). Substituting \(a_1 = 6\) and \(d = 8\), we get \(a_1 = 6\), \(a_n = a_{n - 1}+8\) (for \(n\geq2\)).

Answer:

\(a_1 = \boldsymbol{6}\), \(a_n = \boldsymbol{a_{n - 1}+8}\) (for \(n\geq2\))