Question

which of the following statements is (are) true for all the positive integers?
1) n^3+2n is divisible by 3
2) 5^(2n)-1 is divisible by 24

Answer 1

finally someone looks at it

Answer 2

I think both are correct dude

Answer 3

I just verified them with the help of a calculator

Answer 4

how do you do it on the calculatyor?

Answer 5

i have the ti-84 plus

Answer 6

for 1)
n^3+2n is divisible by 3

let n = 3
3^3 + 2(3) = 27 + 6 = 33 that is divisible by 3

now let us take a big number
n = 24
3^24 +2(24) is divisble by 3

Answer 7

yeah i did that i got 33 then what?

Answer 8

now let us take a big number
n = 24
3^24 +2(24) is divisble by 3 i dont get this?

Answer 9

ohhhhhhhhhhhhhhhhhhh gotcha

Answer 10

swagggg thanks man

Answer 11

ok.. best of luck

Answer 12

Hint: ((n+1)^3+2(n+1)) - (n^3+2n) = 3*(1+n+n^2)

In other words, n^3+2n increases by a multiple of 3 each time you increase n. So, if it's true for n=1 (which it is), it's true for all n.

Similarly, (5^(2(n+1))-1)-(5^(2n)-1) simplifies to 24*25^n, so, when you increase n, you increase 5^(2n)-1 by a multiple of 24.