Sum of a series given the formula for the nth term So I have a series $1+1+3+9+23+\cdots$ and the formula given to find the $n^{th}$ term $R_n=1-2n+2^n$. The first part is to verify that the term $115$ exists in this series. I knew that if I just solve for $n$ in $1-2n+2^n=115$ and if $n$ is an integer solution then the term $115$ exists. In fact I could't solve it and I just guessed and checked that $n=7$. So can someone guide me to solve the following equation?
\begin{align}
1-2n+2^n &= 115\\
-2n+2^n &= 114
\end{align}
Second part of the question is, how can I find the sum of the first 25 terms of this series?
 A: Since
$$ 1-2\cdot n+2^n\ =\ 115 $$
then
$$ 2^n > 115 $$
hence
$$ n > \log_2(115) > 6 $$
Also, by induction,
$$ \forall_{n\ge 4}\quad 2^{n-1} > 2\cdot n-1 $$
Thus,
$$ \forall_{n\ge 4}\quad 2^{n-1}\ < 2^n-2\cdot n + 1
       \ =\ 115 $$
hence (for $\ n\ge 4$)
$$ n-1\ <\ \log_2 115 < 7 $$
i.e.
$$ n\ <\ 8 $$
It follows that IF there is any integer solution $n$
it has t be $\ n=7.\ $ But is there any? The calculation
$$ 2^7-2\cdot 7 + 1\ =\ 115 $$
shows that indeed, $\ n:=7\ $ is a solution, and it is the ONLY integer solution.
REMARK But is there any other REAL solution? -- it would still have to satisfy inequality
$$ 6 < n < 8 $$
But looking at the derivative of $\ 2^x-2\cdot x-1,\ $
which is strictly positive when $\ x>6,\ $ we see that there is no real solution $\ n\ $ but $\ n=7.$
A: The sum is 
2^25[1 + 1/2 +1/4 + 1/8 ..... + 1/2^24] - 2(1+2+3....+25) + 25
This involvss a geometric progression with first term=1 , common ratio = 1/2 and no of terms as 25 
The -ve part is an A.P. with frst term "1"  commn difference = 1 and no.of terms =25
And just add 25 to it 
A: Your guess and check is a good approach when you know $n$ must be an integer.  $2^n$ changes quickly compared with $2n$, so just ignore the $2n$ and find the $n$ which is close.  You can prove that $2^n-2n$ is increasing with $n$ for $n \gt 2$, so if you miss you can just increment $n$ by $1$ and have a proof that there is no $n$.  
The $1-2n$ part is an arithmetic series and the $2^n$ part is geometric.  Sum each and add them together.
