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2n < (n + 2)! for all natural number n.

 

Answers (1)

2n < (n + 2)! .....(given)

 Now, we will substitute different values for n,

P(0) \rightarrow 0 < 2!

P(1) \rightarrow 2 < 3!

P(2) \rightarrow 4 < 4! 

P(3) \rightarrow 6 < 5!

Now, let us consider,

P(k) = 2k < (k+2)! To be true,

Thus, P(k+1) = 2(k+1)< [(k+1) + 2]!

Now, we know that,[(k+1) + 2]! = (k+3)! = [(k+3) (k+2)(k+1) ... 3 \times 2\times 1]

But also,         = 2(k+1) \times (k +3)(k+2) ... 3 \times 1 > 2(k+1)

Thus,2(k+1) < [(k+1) + 2]!

Thus, P(k+1) is true.

Hence, by mathematical induction,

For each natural no. n it is true that, P(n) = 2n < (n + 2)!

Posted by

infoexpert21

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