1 3 2 3 3 3 n 3 formula

Asked by admin @ 13/08/2021 in Math viewed by 405 People

Use mathematical induction to prove that

1 3 + 2 3 + 3 3 + ... + n 3 = n 2 (n + 1) 2 / 4
for all positive integers n.

Answered by admin @ 13/08/2021

1 3 + 2 3 + 3 3 + ... + n 3 = n 2 (n + 1) 2 / 4

STEP 1: We first show that p (1) is true.

Left Side = 1 3 = 1

Right Side = 1 2 (1 + 1) 2 / 4 = 1

hence p (1) is true.

STEP 2: We now assume that p (k) is true

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

add (k + 1) 3 to both sides

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

factor (k + 1) 2 on the right side

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

set to common denominator and group

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

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

We have started from the statement P(k) and have shown that

1 3 + 2 3 + 3 3 + ... + k 3 + (k + 1) 3 = (k + 1) 2 [ (k + 2) 2 ] / 4

Which is the statement P(k + 1).

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