Let S_nSn be a polygonal sequence built by the Series of the Arithmetic Sequence S_n = Sigma_i^(n)a_i where a_n = {1, 4, 7, 10, 13, ...}, find the generating formula for polygonal sequence S_n?

Let S_n be a polygonal sequence built by the Series of the Arithmetic Sequence given by: a_n = {1, 4, 7, 10, 13, ...}. S_n = Sigma_i^(n)a_i find the generating formula for polygonal sequence S_n?

1 Answer
Mar 27, 2018

S_n = (3n^2-n)/2

Explanation:

Given : an arithmetic sequence with common difference d = 3:

a_n = {1,4,7,13,...} and it's
Series S_n = sum_(i=1)^(i=n) a_n = 1+4+7+13+ cdots+n
Required : S_n for all n in ZZ

Solution:
Strategies: Obtain the S_n from the Sum (series) of the arithmetic sequence

Strategy :
S_n = (a_1 +a_n)/2 n " "(1)
This the area of Trapezoid under the blue sequence...
Now the equation of a arithmetic sequence is:
a_n = a_1 + d(n-1) " " (2)
with d=3
a_n = 3n-2 " " (3)
inserting (3) into (1)
S_n = (1+3n-2)/2n = (3n-1)/2 n = (3n^2-n)/2

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