Question

1. the first 5 terms of a sequence are $a_1 = 11$, $a_2 = 23$, $a_3 = 35$, $a_4 = 47$, $a_5 = 59$. based on this information, create an equation that can be used to find the $n^{th}$ term of the sequence, $a_n$. then, use the equation to find the $39^{th}$ term of the sequence. equation: $a_n = \square$ $a_{39} = \square$

Explanation:

Step1: Identify sequence type

Check differences: \(23 - 11 = 12\), \(35 - 23 = 12\), \(47 - 35 = 12\), \(59 - 47 = 12\). It's an arithmetic sequence with first term \(a_1 = 11\) and common difference \(d = 12\).

Step2: Arithmetic sequence formula

The formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n - 1)d\). Substitute \(a_1 = 11\) and \(d = 12\):
\(a_n = 11 + (n - 1) \times 12\)
Simplify: \(a_n = 11 + 12n - 12 = 12n - 1\).

Step3: Find \(a_{39}\)

Substitute \(n = 39\) into \(a_n = 12n - 1\):
\(a_{39} = 12 \times 39 - 1\)
Calculate: \(12 \times 39 = 468\), so \(a_{39} = 468 - 1 = 467\).

Answer:

Equation: \(a_n = 12n - 1\)
\(a_{39} = 467\)