Question

write an explicit formula for $a_n$, the $n^{\text{th}}$ term of the sequence $16, 7, -2,....$

Explanation:

Step1: Identify sequence type

The sequence \(16, 7, -2, \dots\) is arithmetic. Common difference \(d = 7 - 16 = -9\), \(a_1 = 16\).

Step2: Use arithmetic sequence formula

Arithmetic sequence explicit formula: \(a_n = a_1 + (n - 1)d\). Substitute \(a_1 = 16\), \(d = -9\).
\(a_n = 16 + (n - 1)(-9)\)

Step3: Simplify the formula

Expand: \(a_n = 16 - 9n + 9 = 25 - 9n\)

Answer:

\(a_n = 25 - 9n\)