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Hannibal lecter

Member

Registered: 2016-02-11

Posts: 392

Continuous progressions,

continuous progressions, an arithmetical is one which proceeds by equal intervals

a geometrical, one which advances by unequal and proportionally increasing or
decreasing intervals.

I didn't understand the meaning of that can you please illustrate this in plain English and by numbers or examples?

From web i found : Arithmetic progression is a sequence of numbers where the difference between any two consecutive numbers is the same. For example, series 1,2,3,4 is an arithmetic progression that has a common difference
The difference is 1

I know that but at the top it said, "is one which proceeds by equal intervals."

What is mean these sentence? I don't know what it means by saying proceeds by equal intervals.
From the example above I don't see any intervals I searched a lot I didn't see any arithmetic continuous progressions with intervels only a series and a common difference must of the websites
And does he mean by "proceeds" he mean increasing?
And is arithmetic continuous progressions is the same as Arithmetic sequence? Which I found in https://www.mathsisfun.com/algebra/sequences-sums-arithmetic.html

Last edited by Hannibal lecter (2023-06-23 02:19:05)

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Wisdom is a tree which grows in the heart and fruits on the tongue

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Continuous progressions,

The book you're getting this from is not very helpful for someone who does not have Englishas their first language.  The MIF site uses plain English and is the one I recommend.

In an arithmetic progression the difference between two terms is a constant.  Here are some examples:

3, 6, 9, 12, 15, ....  Here the common difference is 3.

7, 4, 1, -2, -5, -8, .... Here the common difference is -3

0.5, 1.0, 1.5, 2.0, 2.5, ..... Here the common difference is 0.5.

The book you quote from using the term 'equal intervals' instead of 'common difference'. the mean the same.

In a geometric progression if you divide any term by the one before it you get a constant number.  This is refered to as a constant ratio between terms. Here are some examples:

1, 2, 4, 8, 16, 32, ...... Here the common ratio is 2. 

1000, 100, 10, 1, 0.1, 0.01, 0.001, .... Here the common ratio is 0.1 or 1/10. Each term is one tenth of the one before it.

5, 15, 45, 135, 405, .... Here the common ratio is 3.

Yes, when it says 'proceeds' it does mean 'increasing' although remember the terms could be getting smaller rather than larger.

Yes,  'arithmetic continuous progressions' is the same as 'Arithmetic sequence'. You might also find books where it is called an 'arithmetic sequence'.

Hope that helps.

Bob

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