Question

Use mathematical induction to prove that the statement holds for all positive integers. Also, can you label the basis, hypothesis, and induction step in each problem. Thanks

1. 2+4+6+...+2n=n^2+n



2. 8+10+12+...+(2n+6)=n^2+7n

Answer 1

assume true for n=k. Then when n=k+1, we have

2+4+...+2k+(2k+2) = k^2 + k + 2k+2
= k^2 + 2k + 1 + k + 1
= (k+1)^2 + (k+1)

Since true for n=1, true for n=2,3,4...

Similarly,

8+10+...+(2k+6)+(2k+8) = k^2 + 7k + (2k+8)
= k^2 + 2k + 1 + 7k + 7
= (k+1)^2 + 7(k+1)