Question

1. given the recursive formula below, write an explicit formula to represent the arithmetic sequence and simplify it. then use the formula to find the 17th term of the sequence.

$a_n = a_{n - 1} + 4.5$; $a_1 = -7$

equation: $a_n = \square$

$a_{17} = \square$

Explanation:

Step1: Recall arithmetic sequence explicit formula

The explicit formula for an arithmetic sequence is \( a_n = a_1 + (n - 1)d \), where \( a_1 \) is the first term and \( d \) is the common difference.

Step2: Identify \( a_1 \) and \( d \)

Given \( a_1=-7 \) and from the recursive formula \( a_n=a_{n - 1}+4.5 \), the common difference \( d = 4.5 \).

Step3: Substitute into explicit formula

Substitute \( a_1=-7 \) and \( d = 4.5 \) into \( a_n=a_1+(n - 1)d \):
\[

$$\begin{align*} a_n&=-7+(n - 1)\times4.5\\ &=-7 + 4.5n-4.5\\ &=4.5n-11.5 \end{align*}$$

\]

Step4: Find \( a_{17} \)

Substitute \( n = 17 \) into \( a_n=4.5n-11.5 \):
\[

$$\begin{align*} a_{17}&=4.5\times17-11.5\\ &=76.5-11.5\\ &=65 \end{align*}$$

\]

Answer:

Equation: \( a_n = 4.5n - 11.5 \)
\( a_{17}=65 \)