Question

The sum of the first n terms of an arithmetic progression is 4n^2 + 2n. Find its nth term of this AP. a) 2n + 1 b) 2n + 2 c) 3n + 1 d) 3n + 2

Answer 1

Hi,

Explanation:

Let's say a the first term and r the commun difference.

`a+(a+r)+(a+2r)+(a+3r)+(a+4r)+...+(a+(n-1)*r\\\displaystyle =\sum_(i=0)^(n-1) (a+r*i)\\=a*\sum_(i=0)^(n-1) (1) + r*\sum_(i=0)^(n-1) (i)\\=a*n+r*(n*(n-1))/(2) \\=(r*n^2)/(2) -(r*n)/(2)+a*n \\\\=4n^2+2n\\\\(r)/(2) =4\Longrightarrow\ r=8\\\\a-(8)/(2) =2\\\\a-4=2\\\\a=6\\`

`r=8\\a=6\\\\n^(th) \ term=a+(n-1)*r\\\\=6+8*(n-1)\\\\=\boxed{8n-2}\\`