$\frac{1}{2\times3}\times2^2+\frac{2}{3\times4}\times2^3+\frac{3}{4\times5}\times2^4+\cdots+\frac{7}{8\times9}\times2^8+\frac{8}{9\times10}\times2^9?$

Question

Background:

This a a problem from “Eleventh IMC International Mathematics Contest, final round(which held in Singapore)$, 2015$ Standard Six paper.

The problem:

Problem $11:$
Given $2\times2-1=3, 3\times2-4=2, 4\times2-5=3,\cdots, 9\times2-10=8.$
What is the simplified value of $\frac{1}{2\times3}\times2^2+\frac{2}{3\times4}\times2^3+\frac{3}{4\times5}\times2^4+\cdots+\frac{7}{8\times9}\times2^8+\frac{8}{9\times10}\times2^9?$

My attempt:

$\displaystyle\frac{1}{2\times3}\times2^2+\frac{2}{3\times4}\times2^3+\frac{3}{4\times5}\times2^4+\cdots+\frac{7}{8\times9}\times2^8+\frac{8}{9\times10}\times2^9$

$=\displaystyle\frac{1}{2\times(2\times2-1)}\times2^2+\frac{2}{3\times(3\times2-2)}\times2^3+\frac{3}{4\times(4\times2-3)}2^4+\cdots+\frac{7}{8\times(8\times2-7)}\times2^8+\frac{8}{9\times(9\times2-8)}\times2^9$

$=\displaystyle\frac{1}{2\times2\times2-2}\times2^2+\frac{2}{3\times3\times2-3\times2}\times2^3+\frac{3}{4\times4\times2-4\times3}\times2^4+\cdots+\frac{7}{8\times8\times2-8\times7}\times2^8+\frac{8}{9\times9\times2-9\times8}\times2^9$

My attempt 2:( Although I have know the solution of this question and approve the answer, I still want to include another attempts that I tried last week.)

I found that $2, 2^2, 2^3,…, 2^9$ is a geometric series with a common ratio of $2$.

$\text{Let}~~S= \displaystyle\frac{1}{2\times3}\times2^2+\frac{2}{3\times4}\times2^3+\frac{3}{4\times5}\times2^4+\frac{7}{8\times9}\times2^8+\frac{8}{9\times10}\times2^9~~(1)$

$\text{And}~~2S= \displaystyle\frac{1}{2\times3}\times2^3+\frac{2}{3\times4}\times2^4+\frac{3}{4\times5}\times2^5+\cdots+\frac{7}{8\times9}\times2^9+\frac{8}{9\times10}\times2^{10}~~(2)$

We then $(2)-(1),$

$\displaystyle S=\frac{1}{2\times3}\times2^2+\left(\frac{1}{2\times3}-\frac{2}{3\times4}\right)\times2^3+\left(\frac{2}{3\times4}-\frac{3}{4\times5}\right)\times2^4+\cdots+\left(\frac{7}{8\times9}-\frac{8}{9\times10}\right)2^9+\frac{8}{9\times10}\times2^{10}$

$\displaystyle S=\frac{1}{2\times3}+\frac{1\times4-2\times2}{2\times3\times4}\times2^3+\frac{2\times5-3\times3}{3\times4\times5}\times2^4+\cdots+\frac{7\times10-8\times8}{8\times9\times10}\times2^9+\frac{8}{9\times10}\times2^{10}$

And I don’t know how to continue from here too.

Now, I don’t sure how to continue from here, as the calculations a looks very complicated if I want to simplify these expressions. Any hints will be appreciated!

Answer 1

We have \begin{align*} &\frac{1}{2\times3}\times2^2+\frac{2}{3\times4}\times2^3+\frac{3}{4\times5}\times2^4+\cdots+\frac{7}{8\times9}\times2^8+\frac{8}{9\times10}\times2^9 \\ =& 1\times \left(\frac{1}{2}-\frac{1}{3}\right)\times 2^2 + 2\times \left(\frac{1}{3}-\frac{1}{4}\right)\times 2^3 + \cdots + 8\times \left(\frac{1}{9}-\frac{1}{10}\right)\times 2^9\\ =& \frac{1}{2}\times 2\times (2\times 1-0) + \frac{1}{3}\times 2^2 \times (2\times 2-1) + \cdots + \frac{1}{9}\times 2^8\times (2\times 8-7) - \frac{1}{10}\times 2^9\times 8\\ =& \frac{1}{2}\times 2\times 2 + \frac{1}{3}\times 2^2 \times 3 + \cdots + \frac{1}{9}\times 2^8\times 9 - \frac{1}{10}\times 2^9\times 8\\ =& 2 + 2^2+\cdots + 2^8 - \frac{1}{10}\times 2^9\times 8\\ =& 2^9-2 - \frac{4}{5}\times 2^9\\ =& \frac{1}{5}\times 2^9-2\\ =& 100.4 \end{align*}