Question

how to prove that the sequence defined by (2n + 3) / (n + 3) is always less or equal than 2, for every n? ( without taking the sequence limit? Thanks

Answer 1

can we induce it?

Answer 2

2n + 3 / n + 3 = 2
Therefore it simplifies to
2n + 3 = 2n + 6 which is noot possible
Therefore n cannot be 2
If 2n + 3 / n + 3 = 3 ( >2)
2n + 3 = 3n + 9
n = -6
If 2n + 3 / n + 3 = 1 ( <2)
2n + 3 = n + 3
n = 0
So i think it is all possible except n = 2

Answer 3

2n + 3
-------
n + 3

-3 | 2 3
0 -6
-----
2 -3



(3)
2 - -----
n+3

Answer 4

3/n+3 >2

3/2 > n+3

1.5 - 3 > n

-1.5 > n

Answer 5

since a sequence starts at 1 and moves up; then n< -1.5 is not possible is it?

Answer 6

Hmm...induction;

(2n+3)/n+3 <=2

Proving this for the first natural number;
5/4<=2 [yes]

Lets prove this is true
2(n+1)+3 / n+1+3 <=2
2n+5 <= 2n + 8
5<=8 [true]

Done!