# factoring/simplification help

#### tehdave

I am trying to factor/simplify the following expression for a proof I am working on, but I am unsure how to do it.
The expression is
$$\displaystyle \frac{a(1-q^n)}{(1-q)}+(a \times q^n)$$

I need the expression to factor/simplify to
$$\displaystyle \frac{a(1-q^{n+1})} {1-q}$$

Any help with the approach I should take is greatly appreciated.
Thanks!

#### papex

yes, the expression simplifies to a(1-q^(n+1))/(1-q)
you can write it: (a(1-q^n)+aq^n(1-q))/(1-q) = (a-aq^(n+1))/(1-q)
=a(1-q^(n+1))/(1-q)

#### Pim

Key steps here are turning the second bit into a fraction, by multiplying by $$\displaystyle \frac{1-q}{1-q}$$
It's also important to realise that $$\displaystyle q*q^n = q^{n+1}$$

$$\displaystyle \frac{a(1-q^n)}{1-q} + (a*q^n)$$
$$\displaystyle \frac{a(1-q^n)}{1-q} +\frac{(1-q)*a*q^n}{1-q}$$
$$\displaystyle \frac{a(1-q^n)}{1-q} +\frac{a*(q^n-q^{n+1})}{1-q}$$
$$\displaystyle \frac{a(1 - q^n+q^n-q^{n+1})}{1-q}$$
$$\displaystyle \frac{a(1-q^{n+1})}{1-q}$$

Do you understand how I did it?

• HallsofIvy