Has anyone ever done this before?

So I'm the type of person that likes doing odd math in his head as he falls asleep and for some reason last night I thought; "what happens if you divide each number by the next in line." I.E. 2/1 3/2 4/3 and so on. This was what I found.

2/1=2
3/2=1.5
4/3=1.333333333r
5/4=1.25
6/5=1.2
7/6=1.1666666666r
8/7=1.142857 loop
9/8=1.125
10/9=1.1111111111r
11/10=1.1
12/11=1.09r
13/12=1.08333333r
14/13=1.076923 loop
15/14=1.0714285 loop
16/15=1.066666r

I things backwards in my head some times so maybe this doesn't really mean anything, but I thought it was oddly cool how the decimal got smaller the higher in count you went.

I don't know really just something I wanted to share and see if anyone else ever tried or could explain it.

 

First you are NOT "dividing each number by the next in line", you are dividing each number INTO the next in line.

So (n+ 1)/n= 1+ 1/n. As n gets larger, 1/n gets smaller so the limit as n goes to infinity is 1

"Dividing each number by the next in line", 1/2, 2/3, 3/4: n/(n+ 1) 1/(1+ 1/n) so the limit is also 1 but the sequence approaches 1 from below while (n+ 1)/n approaches 1 from above.

 

What you're observing is a fascinating mathematical pattern! When you divide each number by the next consecutive number, you're essentially generating a sequence of ratios. What you've noticed is that as the numerator increases and the denominator stays relatively close, the ratio tends to decrease. This pattern is related to the concept of limits in calculus.

As you've seen, the decimal representation of these ratios becomes smaller and approaches a certain value as you move along the sequence. This value is known as the limit of the sequence. In this case, the limit seems to approach 1 as the numbers in the sequence increase.

This phenomenon is a glimpse into the world of mathematical analysis, where understanding the behavior of functions and sequences is crucial. Your observation highlights the beauty and curiosity that mathematics can offer, even in seemingly simple exercises like dividing consecutive numbers.

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