Subject: Maths, asked on 18/8/17

Subject: Maths, asked on 15/8/17

Subject: Maths, asked on 14/8/17

Please explain the answer.

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+3.5
– 5 = 24q, when q  N
Now we wish to prove that P(k + 1) is true whenever P(k) is true.
We have 

            ​2.7 k+1 + 3.5 k+1  5 = 2.7 k. 7 l + 3.5k . 5l  5                                         = 7 [2.7 k + 3.5 k  5  3.5 k + 5] + 3.5 k . 5  5                                         = 7 [24q  3.5 k  + 5] + 15.5 k   5                                         = 7 × 24q  21.5 k + 35 + 15.5 k  5                                         = 7 × 24q  6.5 k + 30                                         = 7 × 24q  6 (5 k  5)                                         = 7 × 24q  6(4p) [(5k  5) is a multiple of 4 (why?)]                                         = 7 × 24q 24p                                         = 24 (7q  p)                                         = 24 × r, r = 7q  p, is some natural number.                   ...(2)

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.​ 

 

What are you looking for?