Let sn sigmak1n 3k 2 so s1 1 s2 1 4 s3 1 4 7 etc Wri

Let s_n = sigma_k=1^n (3k - 2), so s_1 = 1, s_2 = 1 + 4, s_3 = 1 + 4 + 7, etc. Write a (non-homogeneous) recurrence relation for the sequence {s_n}, and find the unique solution to that recurrence meeting the appropriate initial conditions.

Solution

s1 = 1 , s2 = 1 + 4 , s3 = 1 + 4 + 7

every other term is sum of previous term + 3 times the term + 1

so the recurrence relation of the sequence is

so Sn+1 = Sn + 3n + 1

where Sn = 1

so if we have to find S3

plug n = 2

S 2 + 1 = S2 + 3 (2) + 1

S3 = 1+ 4 + 7