Assertion (A) : $a, b, c$ are in A.P. if and only if $2 b=a+c$.\ Reason (R) : The sum of first $n$

Question

Assertion (A) : $a, b, c$ are in A.P. if and only if $2 b=a+c$.\ Reason (R) : The sum of first $n$ odd natural numbers is $n^{2}$.\

Professor PC Thomas Classes · Answer

(b) : Since, $a, b, c$ are in A. P., then $b-a=c-b$\ $\Rightarrow 2 b=a+c$\ First $n$ odd natural numbers be $1,3,5, \ldots(2 n-1)$.\ which form an A.P. with $a=1$ and $d=2$\ Sum of first $n$ odd natural numbers $=\frac{n}{2}[2 a+(n-1) d]$ $$ =\frac{n}{2}[2+(n-1) 2]=n^{2} $$ Hence, both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).