If P(n) is the statement “n(n + 1) is even”, then what is P(3)?
2.7n + 3.5n – 5 is divisible by 24 for all n ϵ N
Let P(n) = 2.7n + 3.5n – 5
Now, P(n): 2.7n + 3.5n – 5 is divisible by 24 for all n ϵ N
Step1:
P(1) = 2.7 + 3.5 – 5 = 1.2
Thus, P(1) is divisible by 24
Step2:
Let, P(m) be divisible by 24
Then, 2.7m + 3.5m – 5 = 24λ, where λ ϵ N.
Now, we need to show that P(m+1) is true whenever P(m) is true.
So, P(m+1) = 2.7m+1 + 3.5m+1 – 5
= 2.7m+1 + 5.( 2.7m + 3.5m – 5 ) – 5
= 2.7m+1 + 5.( 24λ + 5 - 2.7m ) – 5
= 2.7m+1 + 120λ + 25 - 10.7m – 5
= 2.7m.7 - 10.7m+ 120 λ +24 – 4
= 7m(14 – 10) + 120 λ +24 – 4
= 7m(4) + 120 λ +24 – 4
= 7m(4) + 120 λ +24 – 4
= 4(7m - 1) + 24(5λ +1)
As, 7m – 1 is a multiple of 6 for all m ϵ N.
So, P(m+1) = 4.6μ +24(5λ +1)
= 24(μ +5λ +1)
Thus, P(m+1) is true.
So, by the principle of mathematical induction, P(n) is true for all n ϵN.