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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 $n^{th}$ term of an arithmetic sequence and you know the common difference , $d$ , you can find the ${(n + 1)}^{th}$ term using the recursive formula $a_{n + 1} = a_{n} + d$ .

Example 1:

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

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

If you know the $n^{th}$ term and the common ratio , $r$ , of a geometric sequence , you can find the ${(n + 1)}^{th}$ 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 $r = 2$ .

$\begin{array}{l} a_{1} = 3 \\ a_{2} = a_{1} r = 3 (2) = 6 \\ a_{3} = a_{2} r = 6 (2) = 12 \\ a_{4} = a_{3} r = 12 (2) = 24 \end{array}$