Question

The sum of terms equidistant from the beginning and end in an $A.P.$ is equal to ______.

Answer 1

Let $a$ is first term & $d$ is the common difference of the $A.P.$

$\Rightarrow a_r=r^{th}$ term from beginning

$\Rightarrow a_r=a+(r-1)d$ ...$\left(i\right)$

$\therefore a_{r^{\prime }}=r^{th}$ term from end

$\Rightarrow a_{r^{\prime }}=(a+(n-1\left)d\right)+(r-1)(-d)$ ...$\left(ii\right)$

$a_r+a_{r^{\prime }}=a+(r-1)d+(a+(n-1\left)d\right)+(r-1)(-d)$

$\Rightarrow a_r+a_{r^{\prime }}=a+[a+(n-1\left)d\right]$

Therefore, sum of terms equidistant from the beginning and end is always equal to the sum of first and last term of the $A.P.$