Question

State true or false.

$1^3+3^3+5^3+...+{\left(2n-1\right)}^3=n^2\left(2n^2-1\right)$ n is a natural number

• A. True
• B. False

Answer 1

The correct option is A True

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

The result is true for $n=1$

$2n^4-n^2=2{\left(1\right)}^4-{\left(1\right)}^2=2-1=1$

Let the result be true for $n=k$. That is

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

Now we need to prove that the result is also true for $n=k+1$. That is

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

$=2k^4-k^2+{(2k+1)}^3$

$=2k^4-k^2+8k^3+3\times 4k^2+3\times 2k+1$

$=2k^4+8k^3+12k^2+8k+2-2k-1-k^2$

$=2(k^4+4k^3+6k^2+4k+1)-(k^2+2k+1)$

$=2{(k+1)}^4-{(k+1)}^2$

$\therefore$The result is also true for $n=k+1$.

Hence by the principle of mathematical induction the result is true for all $n\in N$