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Question

By principle of mathematical Induction show that an​​​​​​-b is divisible by a+b when n is a positive integer. 


Solution

 Using the principle of mathematical induction, prove that (xn - yn) is divisible by (x - y)for all n ∈ N. Solution:

Let the given statement be P(n). Then, 

P(n): (xn - yn) is divisible by (x - y). 

When n = 1, the given statement becomes: (x1 - y1) is divisible by (x - y), which is clearly true. 

Therefore P(1) is true. 

Let p(k) be true. Then, 

P(k): xk - yk is divisible by (x-y). 

Now, xk + 1 - yk + 1 = xk + 1 - xky - yk + 1

                              [on adding and subtracting x)ky] 

= xk(x - y) + y(xk - yk), which is divisible by (x - y) [using (i)] 

⇒ P(k + 1): xk + 1 - yk + 1is divisible by (x - y) 

⇒ P(k + 1) is true, whenever P(k) is true. 

Thus, P(1) is true and P(k + 1) is true, whenever P(k) is true. 

Hence, by the Principal of Mathematical Induction, P(n) is true for all n ∈ N. 
 

Mathematics

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