Question

What is the sum of 1 cube plus 2 cube plus 3 cubeso on plus n cube is equal to

Answer 1

Answer:

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

Proof by mathematical induction.

When n = 1 the result is clear, 13 = 12

Assume the result is true for n = k, that is

13 + 23 + 33 + 43 + ... k3 = (1 + 2 + 3 +...k)2

Let n = k + 1, then

13 + 23 + 33 + 43 + ... n3

= 13 + 23 + 33 + 43 + ... + (k+1)3

= (1 + 2 + 3 + ... + k)2 + (k+1)3

= (k(k+1)/2)2 + (k+1)3

= (k2 (k+1)2)/4 + (k+1)3

= (k+1)2/4 (k2 + 4k + 4)

= ((k+1)2 (k+2)2)/4

= ((k+1)(k+2)/2)2

= (1 + 2 + 3 + ... + (k+1))2

Thus 13 + 23 + 33 + 43 + ... n3 = (1+2+3+...n)2 for all positive intergers n.

Answer 2

Answer:

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