How does a simplification like this work?

Question

This is an example of mathematical induction from my textbook.

What I want to know is how the equation gets simplified to that. It's probably something simple but I just couldn't figure it out. Thanks!

$$ \begin{align} \frac{a(1+r^k)}{(1-r)} + ar^k &= \frac{a}{1-r}\cdot(1-r^k+r^k-r^{k-1}) \end{align} $$

Are you dealing with https://math.stackexchange.com/questions/658992/proving-the-geometric-sum-formula-by-induction — lab bhattacharjee, Jul 16 '17 at 10:18

score 1 · Accepted Answer · answered Jul 16 '17 at 10:17

1

$$\frac{a(1+r^k)}{1-r}+ar^k=\frac{a(1+r^k)+ar^k(1-r)}{1-r}=\frac{a}{1-r}(1+r^k+r^k-r^{k+1})$$

Maybe you mean the following? $$\frac{a(1-r^k)}{1-r}+ar^k=\frac{a(1-r^k)+ar^k(1-r)}{1-r}=\frac{a}{1-r}(1-r^k+r^k-r^{k+1})$$

answered Jul 16 '17 at 10:17

Michael Rozenberg

Aw man thank you so much I totally forgotten about multiplying with the denominator. – Chan Jing Hong Jul 16 '17 at 10:40

1 Answers1