Question

Prove by mathematical induction that 7^n + 11^n is divisible by n if n is an odd

Answer 1

we can prove this for 1..
now, let

7^k + 11^k = kq be true for some integer q and k = 2m+1

now, for k+2 ( note that we are already concerned about consecutive odd integers)

we have..
7^(k+2) + 11^(k+2) on LHS
=> 7^k .49 + 11^k . 121
=>7^k .49 + 11^k . 49 - 11^k .49 + 11^k .121
=>49(11^k + 7^k) + 11^k (121 -49)
=>49(kq) + 11^k (72)

you sure question is right? i dont see it being represented at k * something..

Answer 2

i just checked for k=5 , as evident from last digit sum(i.e 8) ,,clearly its not divisible by 5..
please recheck the question..

Answer 3

Sorry it is for divisible by 9 not 'n'

Answer 4

i see..

so now let 7^k + 11^k = 9q to be true for some integer q and k = 2m+1

now, for k+2 ( note that we are already concerned about consecutive odd integers)

we have..
7^(k+2) + 11^(k+2) on LHS
=> 7^k .49 + 11^k . 121
=>7^k .49 + 11^k . 49 - 11^k .49 + 11^k .121
=>49(11^k + 7^k) + 11^k (121 -49)
=>49(9q) + 11^k (72)
=>9(49q + 8.72^k)

=>9*(some integer).
hence its true by induction..hope that helped..

Answer 5

Thank you so much, that really helped!

Answer 6

glad to help ^_^

Answer 7

I guess a little error at the last step:
It should be 9(49q + 8*11^k)
^
not 72

Answer 8

yep..the very same..hmm,,