Question

A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying Sk+1 completely.

Sn: 1 + 4 + 7 + . . . + (3n - 2) = n(3n - 1)/2


please, can you help me I need the formula how to solve it.

Answer 1

well, just plug in k and k+1 for n.

You need to prove that if S_k is true, then S_k+1 is also true. Then, if you can show that S1 is true, then S2,S3,... are all true as well.

1+4+...+(3k-2)+(3(k+1)-2) = S_k+3(k+1)-2
= k(3k-1)/2 + 3(k+1)-2
= (k(3k-1) + 6(k+1)-4)/2
= (3k^2-k+6k+6-4)/2
= (3k^2+5k+2)/2
= (k+1)(3k+2)/2
= (k+1)(3(k+1)-1)/2
= S_k+1

Answer 2

thank you