use induction to show that 1^3+3^3+5^3+...+(2n-1)^3=n^2(2n-1)

Answer 1

The correct formula for the sum of the first n cubes, 1³+2³+...+ n ³ = ( n ( n +1)/2)²

the statement is true for n=1, since 1^3 = 1 = (1*(1+1)/2)^2

the induction hypothesis is 1³+2³+...+ n ³ = ( n ( n +1)/2)²

1³+2³+...+ n ³ + (n+1)³  = ( n ( n +1)/2)² + (n+1)³

(n²( n +1)²/4 + (n+1)³=

 (n +1)² (n²/4+n+1)²=

= (n +1)² ((n² +4n+4)/4)²

 =(n +1)² ((n +2/2)²

=((n +1) (n +2)/2)²