Question: An arithmetic sequence has a common difference of $1$ and a geometric sequence has a common ratio of $3$. A new sequence is formed by adding corresponding terms of these two progressions. It is given that the second and fourth term of the sequence are $12$ and $86$ respectively. Find, in terms of $n$,
- the $n^{th}$ term,
- the sum of the first $n$ terms
of the new sequence.
Answers:
- $n+1+3^n$.
- $\frac 1 2 n(n+3) + \frac{3}{2} 3^n - \frac{3}{2}$.
Working so far
I am stuck getting different values for $a$.