How can I simplify this rational expression? I am trying to simplify $$ \frac{3(5^{k+1} - 1) + 4(3 \;\cdot\;5^{k+1})}{4}$$ to get to $$\frac{3(5^{k+2}-1)}{4} $$
I start with $$ \frac{3(5^{k+1} - 1) + 12\;\cdot\;5^{k+1}}{4}$$
$$= \frac{3\;\cdot\;5^{k+1} - 3 + 12\;\cdot\;5^{k+1}}{4} $$
then I am stuck. 

Continuing from where you left off:
$$ \frac{3\;\cdot\;5^{k+1} - 3 + 12\;\cdot\;5^{k+1}}{4} $$
Considering the numerator:
$$3\;\cdot\;5^{k+1} + 12\;\cdot\;5^{k+1} -3$$
$$5^{k+1}(3+12)-3=$$
$$5^{k+1}(15)-3=$$
$$5^{k+1}(5.3)-3=$$
$$5^{k+2}(3)-3=$$
$$3(5^{k+2}-1)$$
So your original expressing is:
$$\frac{3(5^{k+2}-1)}{4}$$
