Question

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.

Answer 1

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).

Answer 2

Answer:

Step-by-step explanation:

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