Question

1. the first 5 terms of a sequence are $a_1 = -0.1$, $a_2 = -0.12$, $a_3 = -0.14$, $a_4 = -0.16$, $a_5 = -0.18$. 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 $24^{th}$ term of the sequence. equation: $a_n = \square$ $a_{24} = \square$

Explanation:

Step1: Identify the sequence type

The sequence has a common difference. Let's check: \(a_2 - a_1=-0.12 - (-0.1)=-0.02\), \(a_3 - a_2=-0.14 - (-0.12)=-0.02\), so it's an arithmetic sequence with first term \(a_1 = -0.1\) and common difference \(d=-0.02\).

Step2: Arithmetic sequence formula

The formula for the \(n\)-th term of an arithmetic sequence is \(a_n=a_1+(n - 1)d\). Substitute \(a_1=-0.1\) and \(d = -0.02\):
\(a_n=-0.1+(n - 1)(-0.02)\)
Simplify: \(a_n=-0.1-0.02n + 0.02=-0.08-0.02n\) or \(a_n=-0.02n-0.08\) (we can also write it as \(a_n=-0.1+(n - 1)(-0.02)\) which simplifies to \(a_n=-0.02n - 0.08\))

Step3: Find \(a_{24}\)

Substitute \(n = 24\) into the formula \(a_n=-0.02n-0.08\):
\(a_{24}=-0.02(24)-0.08\)
\(a_{24}=-0.48-0.08=-0.56\)

Answer:

Equation: \(a_n=-0.02n - 0.08\) (or equivalent like \(a_n=-0.1+(n - 1)(-0.02)\))
\(a_{24}=-0.56\)