Question

If $S$ and $T$ are respectively the sum and the sum of the squares of n successive positive integers beginning with ${}^{\prime }{a}^{\prime }$ then show that $nT-S^2$

• A. Independent of $a$
• B. Dependent of $a$
• C. Varies from condition to condition
• D. None of these

Answer 1

The correct option is A Independent of $a$

According to question

S=a+a+1+a+2+a+3+......+(a+n-1)

$=na+\frac{n(n-1)}{2}-\left(1\right)$

$T=a^2+(a+1{)}^2+....+(a+n-1{)}^2$

$T=n{a}^2+\frac{2an(n-1)}{2}+\sum N^2$'

$nT=n^2{a}^2+a{n}^2(n-1)+n\sum N^2$

and $S^2=n^2{a}^2+\frac{n^2(n-1{)}^2}{4}+a{n}^2(n-1)$

$nT-S^2=n\sum N^2-\frac{n^2(n-1{)}^2}{4}$

Clearly $nT-S^2$ is independent of a.