Can you tell me an easy way to understand “finding a term of a recursive sequence”. Here is the example they have given me, if f(1)=160 and f(n+1)=-2f(n) what is f(4)?

In a sequence, basically the nth term (or just n) is actually the term to be determined (or calculated). Hence, f(1) means, first term, f(2) means second term, and so on.

For this sequence, we have been given the first term, that is;

[tex]f(1)=160[/tex]

To determine the second term we simply substitute for n = 2, that is;

[tex]f(n+1)=-2f(n)[/tex]

However, for this formula given, the left side already already has the nth term plus 1. Therefore, in this case we simply substitute n = 1, and we have the second term. This is shown below;

[tex]f(1+1)=-2f(1)[/tex][tex]f(2)=-2(160)[/tex][tex]f(2)=-320[/tex]

The same way we can now calculate the 3rd and 4th term. Note that the common ratio is -2.

We now have;

[tex]f(2+1)=-2f(2)[/tex][tex]f(3)=-2(-320)[/tex][tex]f(3)=640[/tex]

And to calculate f(4), the fourth term;

[tex]f(3+1)=-2f(3)[/tex][tex]f(4)=-2(640)[/tex][tex]f(4)=-1280[/tex]

ANSWER:

[tex]f(4)=-1280[/tex]