Summation and Arithmetic progression problem

This is the question which I am referring to

If $S_n=an^2 + bn $ , verify that the series $\sum {t_{n}}$ is arithmetic where $ S_n=\sum_{n=1}^{n}{t_n} $

My try:

• first of all I used below equation to calculate ${t_n}$

${t_n} = S_n - S_{n-1} = a (2n-1)-b$

• Then I calculated common difference by below equation

  
  
  

  $d=t_n-t_{n-1}$



• Then I calculated $t_1$ by putting $n=1$.



• Then I calculated $s_n$ by AP formula and it came out to be
  $s_n= an^2-bn$ but for $\sum {t_{n}}$ to be arithmetic $S_n=s_n$

Please mention where am I wrong

Answer

$t_n=a\{n^2-(n-1)^2\}+b\{n-(n-1)\}=2na-a+b$

$$t_n-t_{n-1}=2na-a+b-\{2(n-1)a-a+b\}=2a$$ which being independent of $n$ is constant