Prove that for any value of n >= 1, 8^n - 3^n is a product of 5.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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It has to be shown that `8^n - 3^n` is a product of 5 for all values of n >= 1.

For the base case n = 1, `8^1 - 3^1 = 8 - 3 = 5` which is a product of 5.

Assume `8^n - 3^n` is a product of 5 or `8^n - 3^n = 5*k`

`8^(n +1) - 3^(n + 1)`

=> `8*8^n - 3*3^n`

=> `8*8^n - 3*8^n + 3*8^n - 3*3^n`

=> `8^n(8 - 3) + 3*(8^n - 3^n)`

=> `8^n*5 + 3*5*k`

=> `5(8^n + 3k)`

This is a product of 5

By mathematical induction it has been shown that `8^n - 3^n` is a product of 5 for n = 1 and if it is assumed that `8^n - 3^n` is a product of 5 it follows that `8^(n+1) - 3^(n+1)` is also a product of 5.

This proves that for any value of `n >= 1` , `8^n - 3^n` is a product of 5

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