Answer:
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
