Respuesta :

druiz1518
or n=1, we have 1-1=0, 0 is divisible by 3.
Assume that this is true for n and we have to show that its true for n+1
(n+1)3-(n+1)=n3+3n2+2n=n3-n+3(n2+n) is divisible by 3; n3-n by 3 by is divisible by assumption and 3(n2+n) is divisible as well. So, if n3-n is divisible by 3, then (n+1)3-(n+1) is divisible as well.
amna04352

Answer:

Proof by Induction

Step-by-step explanation:

n³ - n

Proof by induction

Let n = 1

1³ - 1 = 1 - 1 = 0

0 is a multiple of 3

Assume true for n = k

k³ - k is a multiple of 3

Prove for n = k + 1

(k + 1)³ - (k + 1)

k³ + 3k² +3k + 1 - k - 1

k³ + 3k² + 2k

k³ + 3k² + 3k - k

(k³ - k) + 3k(k - 1)

Is a multiple of 3 because:

k³ - k was assumed

3k(k - 1) has a factor '3'

Hence the sum is also a multiple of 3

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