Summing Gradually Changing Numbers

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In summary, the conversation discusses triangular numbers and their relationship to arithmetic sequences. The formula for the nth triangular number is n(n+1)/2, and this can be used to determine if a given number is among the list of triangular numbers. The conversation also mentions the general arithmetic sequence formula and its use in finding the sum of a finite sequence.
  • #1
M40
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Hi.

This is the first Topic I've posted at (the very cool) Physics Forums site so go easy on me. My English is very far from perfect so some used terms may not be used correctly... :)

Anyways... I was writing a script for a website and a mathematics question popped up.

The numbers 1, 3, 6, 10, 15, 21... can be received by summing group of numbers where the first is 0 and each subsequent number is greater then the previous with 1.

for example:

1 = 0 + 1
3 = 0 + 1 + 2
6 = 0 + 1 + 2 + 3
10 = 0 + 1 + 2 + 3 + 4

and so on...

My 2 questions:

(0) I'd love to know if it possible to determine if a given number is among the list of numbers that can be received from such summing.

(1) And if the above is too complicated to discuss here - can you tell me the term in mathematics that describes such summing (if such exists) so I can google the topic?

Any help would be fantastic.

Cheers.
 
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  • #3
Perfect. Thank You.
 
  • #4
That's an example of the more general "arithmetic sequence" in which each number is the previous number plus a fixed difference: an+1= an+ d or a1, a1+ d, a1+ 2d, etc. In your example a1= 1 and d= 1.
It can be shown that the sum of a finite sequence of that type is just the average of the first and last terms in the sequence time the number of terms in the sequence. In your case, if you have n terms then the first number is 1 and the last term is n. Their average is (n+1)/2 and so their sum is n(n+1)/2.
 
  • #5
You're not Gauss, that's for sure :-p
 

FAQ: Summing Gradually Changing Numbers

1. What is the concept of "Summing Gradually Changing Numbers"?

Summing gradually changing numbers is the process of adding a series of numbers that change incrementally by a constant value. This can be thought of as adding the terms of an arithmetic sequence or finding the area under a curve on a graph.

2. How is this concept applicable in the real world?

This concept can be applied in various real-world scenarios such as calculating compound interest, analyzing stock market trends, or measuring the growth of a population over time.

3. What are the key components of "Summing Gradually Changing Numbers"?

The key components of this concept include the initial value, the common difference (or rate of change), the number of terms, and the final value. These components are essential for accurately summing gradually changing numbers.

4. What are some common methods for summing gradually changing numbers?

There are several methods for summing gradually changing numbers, including using a formula for arithmetic sequences, using a graph to find the area under a curve, or using a spreadsheet or calculator to calculate the sum.

5. How can understanding "Summing Gradually Changing Numbers" benefit us?

Understanding this concept can help us make predictions and projections based on past trends, make informed decisions based on data analysis, and better understand the relationship between different variables in a changing system.

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