Question

Sum of the series
$1(1)+2(1+3)+3(1+3+5)+4(1+3+5+7)+\dots .+10(1+3+5+7+\dots +19)$ is equal to

• A. 385
• B. 1025
• C. 1125
• D. 2025
• E. 3025

Answer 1

Answer: (E)

$1(1)+2(1+3)+3(1+3+5)+4(1+3+5+7)$
$+\dots +10(1+3+5+7+\dots +19)$
$=1\times 1+2\times 2^2+3\times 3^2+4\times 4^2+\dots +10\times 10^2$
$=1^3+2^3+3^3+4^3+\dots +10^3={\left[\frac{10(10+1)}{2}\right]}^2$
$[\because$ sum of cubes of $n$ natural numbers $=\frac{n(n+1{)}^2}{2}]$
$={\left[\frac{10\times 11}{2}\right]}^2=(55{)}^2=3025$