Sum of series: 9 + 16 + 29 + 54 + 103 + ... The series is neither AP nor GP, so I tried doing some manipulations to get the expression for the general term. 
$S = 9 + 16 + 29 + 54 + 103 + ... T_n$
$S = 0 + 9 + 16 + 29 + 54 + ... T_{n-1} + T_n$
Subtracting the two:
$0 = 9 + 7 + 13 + 25 + 49 + ... (T_n - T_{n-1}) - T_n$
This gives:
$T_n = 9 + 7 + 13 + 25 + 49 + ... (T_n - T_{n-1})$
The expression obtained for  $T_n$ is neither in AP nor in GP.
I repeated the above process with $T_n$ (which gave me a GP this time) and got this expression finally:
$T_n - T_{n-1} = 7 + 6(2^{n-2} - 1) = 6(2^{n-2}) + 1$
How do I proceed from here to get an expression for the general term as well as the sum of the series? 

From $T_n - T_{n-1} =  6(2^{n-2}) + 1$ you have 
$$T_n = T_0 + \sum_{k=1}^n (6(2^{n-2}) + 1) \\
= T_0 + n + 3 (2^n -1)$$
The general approach is:
$$T_n = a+ n + b 2^n$$
so 
$$9 = T_1 = a + 1 + 2 b\\
16 = T_2 = a + 2 + 4b
$$
which gives indeed
$b = 3$ and $a = 2$, hence 
$$T_n = 2+ n + 3 \cdot 2^n$$
The sum is then 
$$
S_n = \sum_{k=1}^n T_k = 2 n + \frac12 n (n+1) + 3 (2^{n+1} -1)
$$ 
