Sum the Powers

Algebra Level 3

$\dfrac{1}{1+2+2^2}+\dfrac{1}{2+2^2+2^3}+\dfrac{1}{2^2+2^3+2^4}+\dfrac{1}{2^3+2^4+2^5}+.... = \dfrac{a}{b}$

For coprime $a$ and $b$ .

Find $a+b$

1 solution

Danish Ahmed
Jul 3, 2015

$S = \dfrac {1}{1 + 2 + 2^2} + \dfrac {1}{2 + 2^2 + 2^3} + \dfrac {1}{2^2 + 2^3 + 2^4} + \dfrac {1}{2^3 + 2^4 + 2^5} + ...$

$S = \dfrac {1}{1 + 2 + 2^2} + \dfrac {1}{2(1 + 2 + 2^2)} + \dfrac {1}{2^2(1 + 2 + 2^2)} + \dfrac {1}{2^3(1 + 2 + 2^2)} + ...$

$S = \dfrac {1}{7} + \frac {1}{2(7)} + \dfrac {1}{2^2(7)} + \dfrac {1}{2^3(7)} + ...$

Using the formula $a + ar +ar^2 +ar^3 +.... = \dfrac{a}{1-r}$ ,

the sum csn be calculated by replacing $a$ with $\dfrac{1}{7}$ and $r$ with $\dfrac{1}{2}$ .

$S = \dfrac{\dfrac{1}{7}}{1-\dfrac{1}{2}} = \dfrac{2}{7}$