Question

If the $6^{th}$ term and ${12}^{th}$ term of an A.P. are $30$ and $75$ respectively, find the sum of its $20$ terms.

• A. $1275$
• B. $1150$
• C. $1140$
• D. $1130$

Answer 1

The correct option is A $1275$

The $nth$ term and sum of n terms of an arithmetic sequence can be calculated by $a_n=a+(n-1)d$ & $S_n=\frac{n}{2}[2a+(n-1\left)d\right]$
Where, n denotes the total number of terms
a = first term
d = common difference
$S_n$ = Sum of n terms

$\therefore$ $a+5d=30$ ($6^{th}$ term)

and $a+11d=75$ (${11}^{th}$ term)

Solving both, we get

$d=7.5$
$a=-7.5$

The sum of $20$ terms:

$=\frac{20}{2}\left(2\right(-7.5)+(20-1\left)7.5\right)$

$=10(17\times 7.5)$

$=1275$