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

Algebra ->  Expressions-with-variables -> 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      Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
To show by induction: 5%5E%28k%2B2%29+-+7%2A5%5Ek+%2B+12 is divisible by 6.
For n = 1: 5%5E%281%2B2%29+-+7%2A5%5E1+%2B+12+=+102+=+6%2A17, and the statement is true for n=1.
Inductive hypothesis: Assume true for some n = k, i.e., let 5%5E%28k%2B2%29+-+7%2A5%5Ek+%2B+12 be true, and show that
5%5E%28k%2B3%29+-+7%2A5%5E%28k%2B1%29+%2B+12 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

5%5E%28k%2B3%29+-+7%2A5%5E%28k%2B1%29+%2B+12
is also divisible by 6, and the statement is proved.

----------------------------------------------------------------------

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