Proof Problem - 5

First, let us see if this is true for $\large n = 1$ ,

$\large \frac {1^2(1 + 1)^2}{4} = \frac {1(2)^2}{4} = \frac {1(4)}{4} = \frac {4}{4} = 1$ , $\large 1^3 = 1$ , so it is true for $\large n = 1$ ,

Now, assume it is true for $\large n = r$ so that,

$\large 1^3 + 2^3 + 3^3 + \dots + r^3 = \frac {r^2(r + 1)^2}{4}$ ,

Then $\large 1^3 + 2^3 + 3^3 + \dots + r^3 + (r + 1)^3 = \frac {r^2(r + 1)^2}{4} + (r + 1)^3$ ,

That equals $\large \frac {r^2(r + 1)^2 + 4(r + 1)^3}{4} = \frac {(r + 1)^2[r^2 + 4(r + 1)]}{4}$ ,

Which equals $\large \frac {(r + 1)^2(r^2 + 4r + 4)}{4} = \frac {(r + 1)^2(r + 2)(r + 2)}{4}$ ,

Finally that equals $\large \frac {(r + 1)^2(r + 2)^2}{4} = \frac {(r + 1)^2(r + 1) + 1)^2}{4}$ .

If it is true for $\large r$ and $\large r + 1$ , it is true for all positive integers. Q.E.D.