It looks like you're given a recursive sequence defined by
and I assume you wish to solve it.
By definition of the n-th term , we have
and so on.
Notice the patterns:
• the exponent of -3 in the coefficient of on the right side is always 1 less than the subscript on the left side
• the sum of terms not involving is a geometric sum of (subscript - 1) terms,
This suggests that the general n-th term is
and with , this becomes
or more compactly in sigma notation,
We can go on to get a closed form for . Let S denote the sum,
Multiply both sides by -3 :
Subtract this from S - lots of terms will cancel:
It follows that the n-th term of the sequence is explicitly given by
Edit: Just noticed your comment. You can get the 5th term by plugging the 4th term back into the recurrence:
Or, using the n-th term we found above,