Let P(n) be the proposition that the sum of the first n positive integers,
13+23+....+n3=(2n(n+1))2 Basis Step
P(1) is true because 13=1=(21(1+1))2.
Inductive Step
For the inductive hypothesis we assume that P(k) holds for an arbitrary positive integer k. That is, we assume that
13+23+....+k3=(2k(k+1))2 Under this assumption, it must be shown that P(k+1) is true, namely, that
13+23+....+k3+(k+1)3=(2(k+1)(k+1+1))2is also true.
When we add k+1 to both sides of the equation in P(k), we obtain
13+23+....+k3+(k+1)3=
=(2k(k+1))2+(k+1)3
=(2k+1)2(k2+4k+4)
=(2k+1)2(k+2)2
=(2(k+1)(k+1+1))2This last equation shows that P(k+1) is true under the assumption that P(k) is true. This completes the inductive step.
We have completed the basis step and the inductive step, so by mathematical induction we know that P(n) is true for all positive integers n. That is, we have proved that
13+23+....+n3=(2n(n+1))2 for all positive integers n.
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