Question

prove by induction that 7^n-4^n-3^n is divisible by 12, when n is a member of positive integers

Answer 1

you mean when n > 1 ?

Answer 2

when n>0

Answer 3

so do u get it? I really don't know how to do this

Answer 4

ok anyway ..

check for n = 1
7^1 - 4^2 - 3^2 = 0
divisible by 12
check for n = 2 :
7^2 - 4^2 - 3^2 = 24
which is divisible by 12

now assume for n=k that
(7^k - 4^k - 3^k) / 12
divisible

so :
no lets check for n+1 :
(7^(k+1) - 4^(k+1) - 3^(k+1)) / 12
7*7k/12 -4*4^k/12 - 3*3^k/12
(3+4)*7^k/12 - 4*4^k/12 - 3*3^k/12
3*7^k/12 + 4*7^k/12 - 4*4^k/12 - 3*3^k/12
(7^k - 3^k) /4 + (7^k - 4^k)/3
now for each we have to check that they are divisible

(7^k - 3^k)/4
k = 1
true
k = 2
true
assume k=t divisible
(7^t - 3^t)/ 4
k= t+ 1
(7*7^t - 3*3^t )/4
(3*7t + 4*7t - 3*3^t) / 4
7^t + 3*(7^t - 3^t)/4 <- divisible by assumption

do the same with : (7^k - 4^k)/3

Answer 5

hope you are still here and sorry that it took me forever

Answer 6

thank you so much

Answer 7

you are welcome
in fact it was really challenging, almost gave up
im happy to find such questions