SOLUTION: Hello this is a mathematical induction question i need help with. show that 5^(k+2)

SOLUTION: Hello this is a mathematical induction question i need help with. show that 5^(k+2) - 7 * 5^k + 12 is divisible by 6 thank you

Question 1183659: Hello this is a mathematical induction question i need help with.
show that 5^(k+2) - 7 * 5^k + 12 is divisible by 6
thank you
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
To show by induction: is divisible by 6.
For n = 1: , and the statement is true for n=1.
Inductive hypothesis: Assume true for some n = k, i.e., let be true, and show that
is also true.
Now .
Since 5^(k+2) - 7*5^k + 12 is divisible by 6 by the inductive hypothesis, and so is 48, it follows that

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

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Addendum: Sans induction, the statement can be proven quite easily.
,
and thus follows directly the conclusion.

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