Question

Sum the series $1^3+3^3+5^3+..........$ to $n$ terms is

• A. $n^2(2n^2-1)$
• B. $n(2n^2-1)$
• C. $n(2n^2+1)$
• D. $n^2(2n^2+1)$

Answer 1

Answer: (A)

$n^2(2n^2-1)$

$n^{th}$ term of the series will be $(2n-1{)}^3$

$1^3+3^3+5^3+.....+(2n-1{)}^3=1^3+2^3+3^3+4^3+.....(2n{)}^3-(2^3+4^3+6^3+......(2n{)}^3)$

$={\left(\frac{2n(2n+1)}{2}\right)}^2-2^3(1^3+2^3+3^3+.....+n^3)$

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

$=4n^4+4n^3+n^2-2n^4-4n^3-2n^2$

$=2n^4-n^2$

$=n^2(2n^2-1)$