Q3 of 66 Page 12

If P(n) is the statement “2n ≥ 3n”, and if P(r) is true, prove that P(r + 1) is true.

Given. P(n) = “2n ≥ 3n” and p(r) is true.


Prove. P(r + 1) is true

we have P(n) = 2n ≥ 3n


Since, P(r) is true So,


= 2r≥ 3r


Now, Multiply both side by 2


= 2.2r≥ 3r.2


= 2r + 1≥ 6r


= 2r + 1≥ 3r + 3r [since 3r>3 = 3r + 3r≥3 + 3r]


Therefore 2r + 1≥ 3(r + 1)


Hence, P(r + 1) is true.


More from this chapter

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1

If P(n) is the statement “n(n + 1) is even”, then what is P(3)?

Given. P(n) = n(n + 1) is even.


Find. P(3) ?

 2

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Given. P(n) = n3 + n is divisible by 3


Find P(3) is true but P(4) is not true

 4

If P(n) is the statement “n2 + n” is even”, and if P(r) is true, then P(r + 1) is true

Given. P(n) = n2 + n is even and P(r) is true.


Prove. P(r + 1) is true

 5

Given an example of a statement P(n) such that it is true for all n ϵ N.