Astatement 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

S_k = k(3k - 1)/2

S_(k + 1) = [(k +1) (3k +2)]/2

Step-by-step explanation:

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

To find Sk, replace all occurrences of n in Sₙ with k.

S_k = 1 + 4 + 7 + … + (3k - 2) = k(3k - 1)/2

To find S_(k+1), replace all occurrences of n in Sₙ with (k +1)

S_(k + 1) = 1 + 4 + 7 + … + [3(k + 1) -2] = (k +1) [3(k + 1) - 1]/2

After distributing the 3s and simplifying, we get

S_k = 1 + 4 + 7 + … + (3k + 1) = (k + 1) (3k + 2)/2