Hello
Can someone explain why multiplying the second formula by 'r' in the derivation of this formula doesn't change things in any way before subtracting?
...or just some background on the formula
Thanks
I guess I should make this formal to explain it better
If we are dealing with finite series we have this
$\displaystyle \displaystyle\sum_{k=0}^n ar^k=s=a+ar+ar^2+...+ar^n$
So that means $\displaystyle sr=ar+ar^2+...+ar^{n+1}$
So now lets subtract the two
$\displaystyle s-sr=(a+ar+ar^2+...+ar^n)-(ar+ar^2+...+ar^{n+1})$
$\displaystyle =a-ar^{n+1}=a(1-r^{n+1})$
So $\displaystyle s-sr=s(1-r)=a(1-r^{n+1})$
And so $\displaystyle s=\frac{a(1-r^{n+1})}{1-r}$
So multiplying by $\displaystyle r$ did change something, we didnt have the $\displaystyle 1$ and we picked up a $\displaystyle r^{n+1}$
Now what if the series is infinite? Just take the limit as n goes to infinity. If $\displaystyle |r|<1$ then the $\displaystyle r^{n+1}$ tends to zero and so the formula becomes $\displaystyle \frac{a}{1-r}$