This problem I could not solve
Use induction to prove that
is divisible to 14
Induction.
First, we verify a "base case". (like n = 1) I'll assume n is a natural number.
base case: (n = 1)
3^(4+2) + 5^(2+1) = 3^7 + 5^3 = 854. (( 854/14 = 61. so we are good here)
induction step:
Suppose that P(n) is true. That is to say, suppose "3^(4n+2) + 5^(2n+1) is divisible by 14)" = "3^(4n+2) + 5^(2n+1) = 14k for some k" (1)
Show P(n+1) is true.
3^(4(n+1)+2) + 5^(2(n+1)+1)
= 3^(4n+6) + 5^(2n+3)
= 3^4 * 3^(4n+2) + 5^2 * 5^(2n+1)
substitute using (1)
= 81 * 3^(4n+2) + 25 * (14k - 3^(4n+2))
= 81 * 3^(4n+2) - 75 *3^(4n+2) + 25*14k
= 6 * 3^(4n+2) + 25*14k
...
I've made a mistake somewhere in my algebra, b/c 6*3^(4n+2) for n=1 is not divisible by 14. sorry I can't find it (sleepy) but you get the idea?