Recursive Sequence

A recursive sequence is a sequence in which terms are defined using one or more previous terms which are given.

If you know the ${n}^{\text{th}}$ term of an arithmetic sequence and you know the common difference , $d$ , you can find the ${\left(n+1\right)}^{\text{th}}$ term using the recursive formula ${a}_{n+1}={a}_{n}+d$ .

Example 1:

Find the ${9}^{\text{th}}$ term of the arithmetic sequence if the common difference is $7$ and the ${8}^{\text{th}}$ term is $51$ .

$\begin{array}{l}{a}_{9}={a}_{8}+d\\ {a}_{9}=51+7=58\end{array}$

If you know the ${n}^{\text{th}}$ term and the common ratio , $r$ , of a geometric sequence , you can find the ${\left(n+1\right)}^{\text{th}}$ term using the recursive formula. ${a}_{n+1}={a}_{n}\cdot r$ .

Example 2:

Write the first four terms of the geometric sequence whose first term is ${a}_{1}=3$ and whose common ratio is $r=2$ .

$\begin{array}{l}{a}_{1}=3\\ {a}_{2}={a}_{1}r=3\left(2\right)=6\\ {a}_{3}={a}_{2}r=6\left(2\right)=12\\ {a}_{4}={a}_{3}r=12\left(2\right)=24\end{array}$