Context:
Consider a sum; $$s_n= \sum_{m=1}^{n} m2^m$$
It has already been established previously that $s_n = f(n)$ where $f(n)=(2n-2)2^n+2$. Using this information find a simple expression for the sum; $$ t_n=n+2(n-1)+4(n-2)+8(n-3)+\ldots +2^{n-1}\times 1$$
My attempt:
Expanding we get $$ t_n=n+2n-2+4n-8+\ldots$$ $$t_n = n(1+2+4+8+\ldots)-(2+8+24+\ldots)$$ $$t_n=n\sum_{i=0}^{n} 2^i-f(n)$$
We see that the sum involves the geometric sequence with common ratio $2$ and first term of $1$ which can be evaluated using the sum of geomertic sequence formula. $$ \sum_{i=0}^{n} 2^i=2^n-1$$
But this is incorrect. When I put this into Symbolab it writes out the first term which is $1$ and then changes the index by $1$ and uses the geometric sum formula. But why must we first write out the first term before applying the sum formula?
