Question

Prove by PMI that 2^5n>3^3n for all n€N

Answer 1

prove

Step-by-step explanation:

2(5n)>3(3n)n=3,5*3=15,3*3=9

Answer 2

2^5n>3^3n is true by PMI.

GIVEN: That all n belongs to the Natural Numbers.
TO PROVE:2^5n>3^3n
SOLUTION: As we know,
Case 1 : where n = 1

2⁵ⁿ > 3³ⁿ

when n = 1

2⁵ > 3³

25 > 9

Hence, the given statement is true for n = 1

Now let us assume that the statement is true for n =k as well.

Case 2: where n = a

2⁵ᵃ > 3³ᵃ                                 -----Eq 1

Now, taking up n = k + 1,

2⁵⁽ᵃ⁺¹⁾ > 3³⁽ᵃ⁺¹⁾

Now, by using Eq 1,

2⁵ᵃ > 3³ᵃ

2⁵ᵃ. 2⁵ > 3³ᵃ . 2⁵

2⁵ᵃ⁺⁵ >  3³ᵃ. 2⁵

2⁵⁽ᵃ⁺¹⁾ >  3³ᵃ. 2⁵ > 3³ᵃ.3³ > 3³ᵃ⁺³ > 3³⁽ᵃ⁺¹⁾....

Hence the statement is true for n = k +a

Therefore, it can be said that

Using PMI, P(n) is true for all n belonging to Natural Numbers.

#SPJ2