Sum of a series given the formula for the nth term

Question

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?

Answer 1

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.

Answer 2

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 $\ n=7.$

Answer 3

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