Question

Let `S_n` 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`?

Answer 1

`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`