Question

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

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

Answer 1

Answer: (E)

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