Question

How do you find the explicit formula for the following arithmetic sequence $123,116,109,102,95,...$?

Answer 1

Step-1: Given Data:

The given arithmetic sequence is:

$123,116,109,102,95,...$

It can be defined explicitly as:

$a_n=a_1+d\left(n-1\right)$
Here, $a_n$ is the ${\text{n}}^{th}$ term, $a_1$ is the first term, and $d$ is the difference between the terms.

Step-2: Finding the difference from the given sequence:

$\begin{array}{rcl}& \Rightarrow & 116-123=-7\\ & \Rightarrow & 109-116=-7\\ & \Rightarrow & 102-109=-7\\ \Rightarrow 95-109& =& -7\end{array}$

Thus, the difference $d=-7$

Then, from the given sequence we can write as

$\begin{array}{rcl}{a}_n& =& 123-7\left(n-1\right)\\ & =& 123-7n+7\\ & \Rightarrow & a_n=130-7n\end{array}$

Hence, the explicit formula for the following arithmetic sequence is $a_n=130-7n$.