Sequence Within A Sequence 5

We have $\begin{aligned} T_1&=T_1 \\ T_2&=T_1+d_1 \\ T_3&=T_1+d_1+d_2 \\ &\vdots \\ T_n&=T_1+d_1+d_2+\ldots+d_{n-1} \end{aligned}$ Add all these equations together, we have $S_n=nT_1+\displaystyle \sum_{i=1}^{n-1}{\sum_{j=1}^i {d_j}}$ From the question we know that $d_1, d_2, \ldots, d_{n-1}$ is a G.P. (Geometric Progression), suppose $\frac{d_k}{d_{k-1}}=R, ~~k=2,3,\ldots,n-1$ , applying the sum of G.P. formula, we have $\sum_{j=1}^i {d_j}=\frac{d_1(R^i-1)}{R-1}$ Hence, $\begin{aligned} \sum_{i=1}^{n-1}{\sum_{j=1}^i {d_j}}&=\sum_{i=1}^{n-1}{\frac{d_1(R^i-1)}{R-1}} \\ &=\frac {d_1}{R-1}\sum_{i=1}^{n-1}{(R^i-1)} \\ &=\frac{d_1}{R-1}\left [\frac {R(R^{n-1}-1)}{R-1}-n+1\right] \\ &=\frac{d_1}{R-1}\left (\frac {R^n-1}{R-1}-n\right) \\ \end{aligned}$ $\therefore S_n=nT_1+\frac{d_1}{R-1}\left (\frac {R^n-1}{R-1}-n\right)$ Now, substitute $n=16$ , $T_1=5$ , $d_1=1$ , $R=2$ , we could get $S_{16}=65599$ .