Question

the sum of first n terms of a certain series is given as 3npower 2-2n.Show that the series is an arithmetic series

Answer 1

$S_n=3n^2-2n\\\\S_{n-1}=3(n-1)^2-2(n-1)=3(n^2-2n+1)-2n+2\\\\=3n^2-6n+3-2n+2=3n^2-8n+5\\\\S_n-S_{n-1}=a_n\\\\a_n=3n^2-2n-(3n^2-8n+5)=3n^2-2n-3n^2+8n-5=6n-5\\-------------------------------\\a_{n+1}=6(n+1)-5=6n+6-5=6n+1\\\\a_{n+1}-a_n=6n+1-(6n-5)=6n+1-6n+5=6=const.$