Use mathematical induction to prove that the statement holds for all positive integers. Also, label the basis, hypothesis, and induction step.

Question

1 + 5 + 9 + … + (4n -3)= n(2n-1)

Answer 1

for k=1: 1 = 1(2-1)
assume for k
for n=k+1,

1+5+...+(4k-3)+(4(k+1)-3) = k(2k-1) + (4(k+1)-3)
= k(2k-1) + (4k+1)
= 2k^2 - k + 4k + 1
= 2k^2 + 3k + 1
= (k+1)(2k+1)
= (k+1)(2(k+1)-1)