Question

Use mathematical induction to prove the property for all positive integers n. [a^n]^5=a^5n

Answer 1

1. first we show it is true for n=1:\[(a^1)^5=(a)^5=a^5\]
2. then we assume its true for n=k to get:\[(a^k)^5=a^{5k}\]
3. then we attempt to show its also true for n=k+1:\[(a^{k+1})^5=(a^k*a)^5=(a^k)^5*(a)^5=(a^k)^5*a^5\]and from 2. above we know \((a^k)^5=a^{5k}\), so we get:\[(a^{k+1})^5=a^{5k}*a^5=a^{5k+5}=a^{5(k+1)}\]
hence we have proved the relation.

Answer 2

thank you :)

Answer 3

yw