Verification for proof by strong induction of $a^n - 1 = (a - 1)(a^{n-1} + a^{n-2} + a^{n-3}+···+a + 1)$
Solution 1:
To prove the induction step, you should prove that $$a^{n+1}-1=(a-1)(a^n+a^{n-1}+\ldots+a+1),$$ given that $$a^n-1=(a-1)(a^{n-1}+a^{n-2}+\ldots+a+1).$$ Of course this follows easily from the fact that $$(a^{n+1}-1)-(a^n-1)=a^{n+1}-a^n=(a-1)a^n.$$