Arithmetic part given by recurrence (a_{n+1}=2n+5). With care on indexing, find explicit form.
Geometric: (b_{n-2}=\dfrac{1}{9}(-3)^n).
The (n^{th}) term of the A.G. sequence is product of corresponding arithmetic and geometric terms. Sum of first three terms follows by direct computation once the first three terms are listed.
(Exact closed form depends on consistent indexing of (a_n) and (b_n) from the given relations.)