How to show that $\left(\frac{y}{y-1}\right)^y = \left( 1 + \frac 1{y-1}\right)^y $

Question

I need to show

\[\left(\frac{y}{y-1}\right)^y = \left( 1 + \frac 1{y-1}\right)^y \]

I cannot understand how they made this step! Can someone explain how this works?

score 10 · Accepted Answer · answered Oct 19 '12 at 15:30

10

We have \[ \frac y{y-1} = \frac{y-1+1}{y-1} = \frac{y-1}{y-1} + \frac 1{y-1} = 1 + \frac 1{y-1}. \]

answered Oct 19 '12 at 15:30

martini

84,101

Thank you very much Martini! Perfect! – Lukas Arvidsson Oct 19 '12 at 15:32

score 6 · Answer 2 · answered Oct 19 '12 at 15:32

6

$$ \frac{y}{y-1} = \frac{y-1+1}{y-1} = \frac{y-1}{y-1} + \frac{1}{y-1} = 1+ \frac{1}{y-1} $$

There's the line of thought I use. You add and subtract one (adds up to zero, so it's allowed), then you rearrange everything to get the final result.

Edit: crud, not quick enough...

answered Oct 19 '12 at 15:32

endolphins

73

Thank you very much, i had totally forgotten about that "method". But now it is clear! – Lukas Arvidsson Oct 19 '12 at 15:33
This is the same @martini did (+1). :) – Mikasa Oct 19 '12 at 16:30

How to show that $\left(\frac{y}{y-1}\right)^y = \left( 1 + \frac 1{y-1}\right)^y $

2 Answers2