factoring/simplification help

May 2010
1
0
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!
 
May 2010
22
10
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

Dec 2008
91
39
The Netherlands
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?
 
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