Question 1183659
To show by induction:  {{{5^(k+2) - 7*5^k + 12}}} is divisible by 6.

For n = 1:   {{{5^(1+2) - 7*5^1 + 12 = 102 = 6*17}}}, and the statement is true for n=1.

Inductive hypothesis:  Assume true for some n = k, i.e.,  let {{{5^(k+2) - 7*5^k + 12}}} be true, and show that 

 {{{5^(k+3) - 7*5^(k+1) + 12}}}  is also true.

Now  {{{5^(k+3) - 7*5^(k+1) + 12 = 5*5^(k+2) - 7*5*5^k + 12

= 5*(5^(k+2) - 7*5^k) + 12 = 5*(5^(k+2) - 7*5^k + 12) -48}}}.

Since 5^(k+2) - 7*5^k + 12 is divisible by 6 by the inductive hypothesis, and so is 48, it follows that 
 
{{{5^(k+3) - 7*5^(k+1) + 12}}}

is also divisible by 6, and the statement is proved.


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Addendum: Sans induction, the statement can be proven quite easily.

{{{5^(k+2) - 7*5^k + 12 = 5^k*(5^2 - 7) + 12 = 18*5^k + 12 = 6*(3*5^k + 2)}}},

and thus follows directly the conclusion.