Use mathematical induction to prove that for all integers n

Use mathematical induction to prove that for all integers n 0, 2^3n 1 is divisible by 7.

Solution

Ans)

f(n) = 2^3n - 1
f(0) = 2^3(0) -1 == 1 - 1
   = 0
   Here 0 is divided by 7 hence true

we have to prove (2^3n - 1) is divisible by 7
   f(n) = (2^3n - 1)/7
   (2^3n - 1) = 7.f(n)
   (2^3n - 1) = 7k (let)

let n = n+1

f(n+1) = 2^3(n+1) - 1
       = 2^3n . 2^3 -1
       = 2^3n . 8 - 1
       Add +8 and -8
       = 2^3n .8 -1 + 8 - 8
       =8(2^3n - 1) + 7
       we have (2^3n - 1) = 7k
       so after substituting we get
       = 8.(7k) - 7
       =7.(8k - 1)
       = 7n
   It is a multiple of 7
   so,n = n+1 is true
   Hence it is proved that (2^3n - 1) is divisible by 7