Question 1044530
Prove that 7^(2n)+16n-1 is divisible by 64
:
we prove this by using mathematical induction
:
For n=1, we have 7^2 +16 -1 = 64 which is divisible by 64
:
For n=k, we assume that 7^(2k)+16k-1 is divisible by 64
:
We must check for n=k+1, is the statement true
:
7^(2(k+1)) +16(k+1) -1 = 49 * 7^2k + 16k +15
:
we adjust this expression by adding and subtracting terms, that is
we add 49*16k, -49 but we have to subtract 49*16k and -49
:
we have the following
:
49*7^2k +16k +15 = 49*7^2k + 49*16k -49 -49*16k -49 +16k +15 =
:
49(7^2k +16k -1) - 49(16k-1) +16k +15 =
:
49(7^k +16k -1) - 48*16k +64
:
all these terms are divisible by 64, so we are done   :-)
: