I could be over thinking or tired... But I am to embarrassed to ask my prof. this probably very simple algebra rule I am ignorant of... Also this is just a snip-it from a inductive proof example.
Say you have something like this
$${\frac{4^k - 1}{3}} +{\frac{3*4^k}{3}} =$$ $${\frac{4*4^k-1}{3}}$$ My question is what algebra rules or rules of fractions allow this? In other words what computation is going on? I can agree that the 3 should cancel out $${\frac{3*4^k}{3} =} {{4^k}}$$ But wouldn't that mean it be something like this $${\frac{(4^k-1)*4^k}{3}}$$ or more accurately $${\frac{(4^k-1)}{3}{* 4^k}}$$ and not this? $${\frac{4*4^k-1}{3}}$$
${\frac{4^k - 1}{3}} +{\frac{3\cdot4^k}{3}} \\ \text{ these fractions have the same denominator so we can write the fraction as one now } \\ \frac{4^k-1+3 \cdot 4^k}{3} \\ \text{ recall } u+3u=4u \\ \text{ so we have } \\ \frac{4^k(1+3)-1}{3}=\frac{4^k (4)-1}{3}=\frac{4 \cdot 4^k-1}{3}=\frac{4^{k+1}-1}{3} \\ $