I am beginning to learn about proofs by induction, but cant get my head around the steps taken in a slide we've been shown.
My problem lies within the red box, and I'm not sure if I'm missing some very basic math here. Particularly, between lines 2 and 3 of the red box:
$$3^{2k} 3^2 \text{ becomes } 8 \cdot 3^{2k}$$
How does this happen? I presume the answer is trivial but I don't want to skip over it.
On
See also rules for exponentiation.
We have $$P(k+1)=3^{2(k+1)}+11$$ $$=3^{2k+2}+11$$ $$=3^{2k}3^{2}+11$$ $$=9\cdot3^{2k}+11$$ $$=(8+1)\cdot3^{2k}+11$$ $$=8\cdot3^{2k}+3^{2k}+11$$ $$=4(2\cdot3^{2k})+3^{2k}+11.$$
$3^2 = 9 = 8 + 1$, so:
$$3^{2k} 3^2 + 11$$ $$= 3^{2k} (8 + 1) + 11 $$ $$= 3^{2k} \cdot 8 + 3^{2k} \cdot 1 + 11 \tag{distributive law}$$ $$= 8 \cdot 3^{2k} + 3^{2k} + 11.$$