Question

Prove that n²+2 cannot be an A.P. for every natural number.

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Answer 1

Answer:

Step-by-step explanation:

Step 1.: Let n=1 ⇒ n<2n holds, since 1<2.

Step 2.: Assume n<2n holds where n=k and k≥1.

Step 3.: Prove n<2n holds for n=k+1 and k≥1 to complete the proof.

k<2k, using step 2.

2×k<2×2k

2k<2k+1(1)

On the other hand, k>1⇒k+1<k+k=2k. Hence k+1<2k(2)

By merging results (1) and (2).

k+1<2k<2k+1

k+1<2k+1

Hence, n<2n holds for all n∈N

Answer 2

Answer:

the answer is

Step-by-step explanation:

Proof by induction.

Let n∈N.

Step 1.: Let n=1 ⇒ n<2n holds, since 1<2.

Step 2.: Assume n<2n holds where n=k and k≥1.

Step 3.: Prove n<2n holds for n=k+1 and k≥1 to complete the proof.

k<2k, using step 2.

2×k<2×2k

2k<2k+1(1)

On the other hand, k>1⇒k+1<k+k=2k. Hence k+1<2k(2)

By merging results (1) and (2).

k+1<2k<2k+1

k+1<2k+1

Hence, n<2n holds for all n∈N