Browse Questions

# Find the sum of the numbers in the $n^{th}$ set: $(1)$, $(2,3)$, $(4,5,6)$, $(7,8,9,10)$.....

$(a)\;\large\frac{n(n^2+2)}{2}\qquad(b)\;\large\frac{n^2(n+1)}{2}\qquad(c)\;\large\frac{n(n^2+1)}{2}\qquad(d)\;None\;of\;these$

Answer : (c) $\;\large\frac{n(n^2+1)}{2}$
Explanation : So , sum of number in $\;n^{th}\;$ set =sum of number of upto $\;n^{th}\;$ set - sum of numbers upto $\;(n-1)^{th}\;set$
$S=S_{\large\frac{n(n+1)}{2}}-S_{\large\frac{(n-1)n}{2}}$
$=\large\frac{(\large\frac{n(n+1)}{2})(\large\frac{n(n+1)}{2}+1)}{2}-\large\frac{(\large\frac{n(n-1)}{2})(\large\frac{n(n-1)}{2}+1)}{2}$
$=n\;(\large\frac{n^2+1}{2})\;.$