Question

The sum of $3^{rd}$ and ${15}^{th}$ elements of an arithmetic progression is equal to the sum of $6^{th},{11}^{th}and{13}^{th}$ elements of the same progression. Then which element of the series should necessarily be equal to zero?

• A. $1st$
• B. $9th$
• C. $12th$
• D. None of the above

Answer 1

The correct option is C $12th$

Let the first term of AP be $a$ and difference be $d$

We know that $n$th term, $a_n=a+(n-1)d$, where $a$ & $d$ are the first term amd common difference of an AP.

$\therefore$ Then third term will be $=a+2d$
${15}^{th}$ will be$=a+14d$
$6^{th}$ will be$=a+5d$
${11}^{th}$ will be$=a+10d$

${13}^{th}$ will be$=a+12d$
Then the equation will be

$a+2d+a+14d=a+5d+a+10d+a+12d$
$2a+16d=3a+27d$
$a+11d=0$
We understand $a+11d$ will be the ${12}^{th}$ term of arithmetic progression.
So, the correct answer is $12$.