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  • For any natural number n, 7n -2n is divisible by 5.

For any natural number n,7^n -2^n is divisible by 5.

Answers (1)

P(n)=7^n -2^n is divisible by 5.        ...............(given)

Now, we will substitute different values for n,

P(0) = 7^0 -2^0 = 0, is divisible by 5

P(1) = 7^1 -2^1 = 5, is divisible by 5

P(2) = 7^2 -2^2 = 45, is divisible by 5

P(3) = 7^3 -2^3 = 335, is divisible by 5

Now, let us consider that,

P(k) = 7^k -2^k be divisible by 5.

Thus, 7^k -2^k=5x

We also get,

P(k+1) = 7^{k+1} -2^{k+1} = (5+2)7^k -2(2)^k

= 5(7)^k + 2(5x), is divisible by 5

Thus, P(k+1) is true

Hence, by mathematical induction,

For each natural no. n it is true that, P(n) = 7^n -2^n is divisible by 5.
 

Posted by

infoexpert21

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