Question

Consider a sequence whose sum to $n$ terms is given by the quadratic function $S_n=3n^2+5n$ The nature of the given series is

• A. A.P
• B. G.P
• C. H.P
• D. AGP

Answer 1

Answer: (A)

A.P

To convert from $S_n$ to $T_n$
we can write $T_n=S_n-S_{n-1}$
Shortcut Method: If $S_n$ is quadratic in $n$
then:Replace $n^2$ by $(2n-1)$ and $n$ by $1$.
Using Shortcut method:$T_n=3(2n-1)+5\left(1\right)$
$=6n-2+5=6n+3$
Thus, $T_n=6n+2$
To check the nature of the series let us check the $T_n$ for the values $n=1,2,3,.$
$T_1=6\times 1+2=6+2=8$
$T_2=6\times 2+2=12+2=14$
$T_3=6\times 3+2=18+2=20$
Thus the sequence is $8,14,20,..$ whose first term is $8$ and the common difference is $14-8=20-14=6$
Hence, the nature of the given series is in A.P