# Solving heterogeneous successions

I know how to get the explicit formula for homogeneous successions, kinda. What I do is get the characteristic equation, get the solutions and then solve a system to obtain the values of A,B,C... constants to build the explicit formula.

... But what if the succession is heterogeneous? Particularly, this question:

Determine the explicit formula for the succession defined by recurrence by $a_n=a_{n-1}+5$ with $a_1=3$

Which, apparently, is heterogeneous.

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Try writing out a few terms and thinking about a formula that might fit the results. – Gerry Myerson Jul 9 '12 at 7:29
This answer illustrates a technique that will work for problems of this kind. – Brian M. Scott Jul 9 '12 at 7:46

If you have that $a_n=a_{n-1}+5$, with $a_1=3$, then you have that $a_n=5(n-1)+3$, because $$a_n=a_{n-1}+5=(a_{n-2}+5)+5=a_{n-2}+10=(a_{n-3}+5)+10=a_{n-3}+15=$$$$...=a_1+5(n-1)$$