Prove by using the principle of mathematical induction 32n – 1 is divisible by 8 for n N.

Asked by Topperlearning User | 4th Jun, 2014, 01:23: PM

Expert Answer:

Let P(n) :  32n – 1 is divisible by 8.
P(1): 32 – 1 = 8 which is divisible by 8. Thus P(n) is true for n = 1.
Let P(k) be true for some natural number k.
i.e.  P(k): 32k – 1 is divisible by 8.
Let 32k – 1 = 8d, where d  N.
Now we prove that P(k + 1) is true whenever P(k) is true.
Now, 32(k + 1) – 1 = 32k.32 – 1
                            = (8d + 1).32 – 1
                            =  9.8d + 9 – 1
                            = 8(9d + 1)
32(k + 1) – 1 is divisible by 8. Thus P(k +1) is true whenever P(k) is true.
By principle mathematical induction 32n – 1 is divisible by 8 for all n  N. 

Answered by  | 4th Jun, 2014, 03:23: PM