Example 6 prove that
2.7n+ 3.5n – 5 is divisible by 24, for all n N.
solution Let the statement p (n) be defined as
P(n) : 2.7n +3.5n – 5 is divisible by 24. We note that P(n) is true for n =1, since 2.7 + 3.5 5 = 24, which is divisible by 24.
Assume that p(k) is true
i.e 2.7k+3.5k – 5 = 24q, when q N
Now we wish to prove that P(k + 1) is true whenever P(k) is true.
The expression on the R.H.S. of (1) is divisible by 24. Thus P(k + 1) is true whenever P(k) true
Hence, by principle of mathematical induction, P(n) is true for all n N.