I want to get ratio of x to y in this equation: $x + 2*(x/10)^2 + 3*(x/10)^3/10 + 4*(x/10)^4/100 + ... + n*(x/10)^n/10^{n-2} = y$

Question

I need the ratio of x to y in this equation:

$\displaystyle \sum ^{\infty }_{n=1} n\frac{\left(\frac{x}{10}\right)^{n}}{10^{n-2}} =y$

or:

x + 2*(x/10)^2 + 3*(x/10)^3/10 + 4*(x/10)^4/100 + ... + n*(x/10)^n/10^(n-2) = y

n → ∞

I expand it like this for better understanding:

1*(x/10)^1/10^(1-2) + 2*(x/10)^2/10^(2-2) + 3*(x/10)^3/10^(3-2) + ... + n*(x/10)^n/10^(n-2) = y

n → ∞

Thanks

Answer 1

$$ n\frac{\left(\frac{x}{10}\right)^n}{10^{n-2}} = n\frac{\left(\frac{x}{10^2}\right)^n}{10^{-2}} =n\left(\frac{x}{10^2}\right)^n\cdot 10^2 $$ so we have $$ \sum_{n=1}^\infty n\left(\frac{x}{10^2}\right)^n\cdot 10^2 =\sum_{n=1}^\infty nz^n\cdot 10^2 = y $$ then we have the fact $$ \sum_{n=1}^\infty nz^n = \frac{z}{(1-z)^2} = \frac{y}{10^2} $$ solve for the fraction.

Answer 2

You have an arithmetico-geometric series. The $n$ is the arithmetic part. The $\frac {\left(\frac x{10}\right)^n}{10^{n-2}}=\frac {x^n}{10^{2n-2}}=100\left(\frac x{100}\right)^n$ is the geometric part.