limits in arithmetic series?

Question

hi..im not very sure if this is the correct term or whatever but this is a math question i have

you see in the sequence:

2/5, 3/7, 4/9
THE PATTERN IS : n+1/2n+3
and no matter what number you plug in as 'n', the limit is always 0.5
because if you simplify this 'n+1/2n+3', the ratio is 0.5

but this rule doesn't seem to work here:

3000000/1 , 3000001/4 , 3000002/7

where the pattern is : n+2999999/3n-2

no matter what 'n' i plug in........the answer says that the limit is always '1/3' , but i can't seem to get that answer!?

Answer 1

in the second case, the sequence of factions will go more slowly towards 1/3 than the first sequence towards 1/2.
In long run n+2999999/3n-2
will approach 1/3.
The problem with the limits is that doesnt matter what happens first. It matters what happens afetr a sufficient amount of time( or of terms)

Answer 2

(n+1)/(2n+3) = (1+1/n)/(2+3/n).The terms which have n in the denominator tend to zero.So the limit 1s 1/2.
The same happens in the secon case
(1+2999999/n)/(3-2/n) => 1/3 .Any term like C/n
with C constant tends to 0 as n=> to infinity

Answer 3

perhaps it's because you are adding 3 million to the numerator
DOH - and you won't be anywhere close -

reread your second problem - there is an error there somewhere