Question

1. in a sequence of numbers, $a_3 = 2$, $a_4 = -2$, $a_5 = -6$, $a_6 = -10$, $a_7 = -14$. based on this information, create an equation that can be used to find the $n^{\text{th}}$ term of the sequence, $a_n$. then, use the equation to find the $22^{\text{nd}}$ term of the sequence. equation: $a_n = \square$ $a_{22} = \square$

Explanation:

Step1: Identify sequence type (Arithmetic)

Check differences: \(a_4 - a_3 = -2 - 2 = -4\), \(a_5 - a_4 = -6 - (-2) = -4\), so common difference \(d = -4\).

Step2: Use arithmetic sequence formula

Arithmetic sequence formula: \(a_n = a_1 + (n - 1)d\). First, find \(a_1\). Using \(a_3 = a_1 + 2d\), substitute \(a_3 = 2\), \(d = -4\):
\(2 = a_1 + 2(-4)\)
\(2 = a_1 - 8\)
\(a_1 = 10\).

Step3: Write \(a_n\) equation

Substitute \(a_1 = 10\), \(d = -4\) into formula:
\(a_n = 10 + (n - 1)(-4)\)
Simplify: \(a_n = 10 - 4n + 4 = 14 - 4n\).

Step4: Find \(a_{22}\)

Substitute \(n = 22\) into \(a_n = 14 - 4n\):
\(a_{22} = 14 - 4(22) = 14 - 88 = -74\).

Answer:

Equation: \(a_n = 14 - 4n\)
\(a_{22} = -74\)