Question

If in a series $S_n=a{n}^2+bn+c$, where $S_n$ denotes the sum of $n$ terms, then

• A. The series is always arithmetic
• B. The series is arithmetic from the second term onwards
• C. The series may or may not be arithmetic
• D. The series cannot be arithmetic

Answer 1

Answer: (B)

$S_n=a{n}^2+bn+c$
$\therefore S_{n-1}=a(n-1{)}^2+b(n-1)+c$ for $n\ge 2$
$\therefore t_n=S_n-S_{n-1}$
$=a\left\{n^2-(n-1{)}^2\right\}+b\{n-(n-1)\}=a(2n-1)+b$
$\therefore t_n=2an+b-a,n\ge 2$
$\therefore t_{n-1}=2a(n-1)+b-a$ for $n\ge 3$
$\therefore t_n-t_{n-1}=2a(n-n+1)=2a$ for $n\ge 3$
$\therefore t_3-t_2=t_4-t_3=..........2a$
Now $t_2-t_1=\left(S_2-S_1\right)-S_1=S_2-2S_1$
$=\left(a\cdot 2^2+b\cdot 2+c\right)-\left\{a\cdot 1^2+b\cdot 1+c\right\}=2a-c\ne 2a$
$\therefore$ Series is arithmetic from the second term onwards.