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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 .

a 9 = a 8 + d a 9 = 51 + 7 = 58

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 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 .

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