Prove that for any natural numbers n, `7^(n)-2^(n)` is divisible by 5.

1.	Prove that for any natural numbers n, `7^(n)-2^(n)` is divisible by 5.
Answer» Consider the given statement is `P(n):7^(n)-2^(n)` is Step I We observe that P(1) is true. `P(1)=7^(1)-2^(1)=5`, which is disivible by 5. Step II Now, assume that P(n) is true for n=k. `P(k)=7^(k)-2^(k)=5q` Step III Now, to prove P(k+1) is true, `P(k+1):7^(k+1)-2^(k+1)`. `=7^(k)2^(k)2` `=7^(k)(5+2)-2^(k)2` `=7^(k)5+27^(k)-2^(k)2` `57^(k)+2(7^(k)-2^(k))` `=5*7^(k)+2(5q)` `=5(7^(k)+2q)`, which is divisible by 5. [from step II] So, P(k+1) is true whenever P(k) is true. Hence, by the principle of mathematical induction P(n) is true for any natural number n.

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