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in Mathematical Induction by (22.8k points)
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If n is a positive integer, then n3 + 2n is divisible by

(a) 2 

(b) 6 

(c) 15 

(d) 3

1 Answer

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Answer: (D) = 3

For n = 1, n3 + 2n = 1 + 2 = 3 which is divisible by 3 and none of the other given alternatives. 

∴ We shall prove n3 + 2n divisible by 3 for all n∈N. 

Let T(n) = n3 + 2n is divisible by 3. 

Basic Step: 

For n = 1, T(1) = n3 + 2n = 1 + 2 = 3 is divisible by 3 is true. 

Induction Step: 

Assume T(k) to be true, i.e., T(k) = k3 + 2k is divisible by 3 

= k3 + 2k = 3m, where m∈N. ...(i) 

Now we need to prove that T(k + 1) holds true, i.e., 

(k + 1)3 + 2(k + 1) is divisible by 3. 

(k + 1)3 + 2(k + 1) = k3 + 3k2 + 3k +1 + 2k + 2

= (k3 + 2k) + (3k2 + 3k + 3) 

= 3m + 3 (k2 + k + 1) (From (i)) 

⇒ T(k + 1) = (k + 1)3 + 2 (k + 1) is divisible by 3, whenever T(k) = k3 + 2k is divisible by 3. 

⇒ n3 + 2n is divisible by 3  ∀ n∈N.

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