Proof by Induction question - help me understand this algebraic manipulation

Question

I'm not sure how we get from this:

$$ \frac{3\cdot5^{k+1}-1}{4} + 3\cdot5^{k+1} $$

to this:

$$ 5^{k+1} \cdot \bigg(\frac{3}{4} + 3\bigg) - \frac{3}{4} $$

Any help understanding this will be greatly appreciated. Thanks!

Answer 1

Just factorize $5^{k+1}$:

${\dfrac{3}{4}(5^{k+1}-1)} + (3*5^{k+1})=(\dfrac{3}{4}+3)5^{k+1}-\dfrac34=(5^{k+1}) * ((3/4) + 3) - (3/4)$

Answer 2

If you are having hard time manipulating it, just make the expression something simpler. For example, define $a = 5^{k+1}$. Then we have $$\frac{3\cdot(a-1)}{4} + 3a = \frac{3a-3}{4}+3a = \frac{3a}{4}-\frac{3}{4}+3a$$ So taking $a$ as the common term, we have $$a\bigg(\frac{3}{4}+3\bigg)-\frac{3}{4}$$ Now since $a = 5^{k+1}$ is defined in the beginning, we can put $5^{k+1}$ instead of $a$ again to get $$5^{k+1}\cdot\bigg(\frac{3}{4}+3\bigg)-\frac{3}{4}$$