Question

which of the following statements is (are) true for all postitive intergers
1) n^3+2n is divided by 3
2)5^(2n)-1 is divided by 24

Answer 1

are you doing induction?

Answer 2

divisible*

Answer 3

yess i think

Answer 4

im pretty sure 1 is true, do you need to prove it?

Answer 5

nope

Answer 6

its on e2020

Answer 7

i have no clue how to do it lol

Answer 8

it is true for the case when n = 1
assume n^3+2n is divisible by 3 for some n in N
now we show the case for n+1
(n+1)^3 +2(n+1)
= (n+1)(n^2+2n+1) + 2n + 2
= (n^3+2n^2+n+n^2 + 2n + 1 + 2n +2)
(n^3+3n^2+5n+3) = (n^3+2n )+ 3(n + 1+n^2)
we assume (n^3+2n) is divisble by 3 and we know 3(n+1+n^2) is divisble by 3
thus by induction n^3+2n is divisible by 3 for all n in N

Answer 9

I think the second one is true also, you can show it by the same method.