problem: math induction 1+27+125... +(2n-1)^3=n^2(2n^2-1)


Please help!!

1+27+125... +(2n-1)^3=n^2(2n^2-1)
this can be expressed in the form :
1^3+3^3+5^3... (2n-1)^3, now you can write the sum of the cubical equation using the formula and conclude the result...

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

Add (2k+1)^{3} to both sides:

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

The right side becomes (k+1)^{2}(2k^{2}+4k+1)=(k+1)^{2}(2(k+1)^{2}-1)

And the proof is complete.