Question

In an arithmetic sequence, $a_{17}$ = -40 and $a_{28}$ = -73. Explain how to use this information to write a recursive formula for this sequence.

Answer 1

The common difference is equal the difference between the given terms divide by the difference between the term numbers:

$d=\dfrac{a_{28}-a_{17}}{28-17}\to d=\dfrac{-73-(-40)}{11}=\dfrac{-33}{11}=-3$

Calculate the first term:

$a_1=a_{17}-16d\\\\a_1=-40-16(-3)=-40+48=8$

The recursive formula is:

$a_n=\left\{\begin{array}{ccc}a_1=8\\a_{n+1}=a_n-3\end{array}\right$