See also airthmeticGeometricSeries.htm

- Sum to n of the Natural Numbers Using Differences
- Sum of the natural numbers using summation
- Sum of Natural Numbers Using Errors
- Second Approach With Errors
- Using Infinite Calculus to find the Sum of the first n Natural Numbers
- Summing the first n natural numbers by finding a general term
- Summing the First n Natural Numbers Using Number Theory
- New Formula for the Sum of the Natural Numbers
- Application - Sum of Odd Numbers
- Application - Sum of Even Numbers
- Application - Sum of Part of the Series of Natural Numbers

For instance, the following table shows the sum of some natural numbers, but we have also used zero, for convenience:

n | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|

S_{n} |
0 | 1 | 3 | 6 | 10 | 15 |

Δ_{1} |
1 | 2 | 3 | 4 | 5 | |

Δ_{2} |
1 | 1 | 1 | 1 |

Because we find that Δ

Using our values, we substitute 0, 1, and 3 in the Equation:

n | Equation | Equation Number |
---|---|---|

0 | c=0 | 1 |

1 | a+b=1 | 2 |

2 | 4a+2b=3 | 3 |

In Equations 2 and 3, we have noted that c=0.

By subtracting twice Equation 2 from Equation 3, we get:

2a=1,

So

a=1/2

Substituting the value for a in Equation 2, we find that b is also 1/2,

So the sum of the first n natural numbers, S

[As a word to the wise, the constant value in the table above is always (n!)a, so in the example, a=1/2!, or 1/2. The sum of the coefficients of the sum of the powers of the natural numbers is always 1.]

That is, the sum of n natural numbers is the sum of n+1 natural numbers less (n+1).

Expanding the term for (k+1), we get:

Which simplifies to:

We can try another approach, and look for the sum of the squares of the first n natural numbers, hoping that this sum will vanish.

Expanding the (k+1)th term:

Expanding (n+1)

Which gives us the sum of the first n natural numbers:

The area of the triangular graph, if we take it as representing n numbers is:

n

So the sum to n terms is, approximately:

If we take E

Looking at the graph, the error on each number is 1/2, so the error on the sum of the first n numbers, E

So:

[3.1]

And the error on S

[3.2]

Noting the difference between the two sum, in preparation for subtracting 3.2 from 3.1:

[3.3]

Subtracting 3.2 from 3.1, taking into account 3.3:

[3.4]

Simplifying and rearranging:

[3.5]

Normally, this would be the start of things, but in this case, the error is simply a number, independent of the actual n-value, so it is the same for all n, and the error is n/2, as we found above.

This approach is similar to the previous one, but introduces a different approach that can be used with other natural number sums.

In the graph below, we have the first few natural numbers shown as rectangles on the graph y=x.

The area under the graph approximates the sum of the natural numbers, so:

[3.21]

With E

[3.22]

Each of our rectangles in the graph have an area k∙1, so our error is k minus the area under the graph between (x=k-1 and x=k):

[3.23]

Working out the integral, we get:

[3.24]

Replacing the integral in 3.23:

[3.25]

Simplifying the sum, adding it up:

So substituting our value for E

Which comes as no surprise!

We note the difference between the sum of the first n natural numbers, and the sum to (n-1) is n

The idea is that we look at the terms S

And for the sum to (n-2):

Continuing to unravel:

And one more:

A pattern becomes clear. So we can write a general term, the k-th term:

[3.3.1]

We have noted that the "n" term is always kn, and the other term is a sum of the natural numbers.

Now we can make k=n, and so get our nth term, noting S

[3.3.2]

The sum above is one n short of S

[3.3.3]

Substituting 3.3.3 in 3.3.2:

Rounding up stragglers:

And finally,

According to number theory, we can represent numbers in one of these four ways:

[4.1]

Where 4m±1 clearly represents the odd numbers.

We try to form an expression for the squares of the odd numbers by squaring 4m±1,

[4.2]

Factorising part of 4.1:

[4.3]

If we form this expression

[4.4]

We can write the square of any odd number as:

[4.5]

We can write an odd number as (2n-1), where n is a natural number.

n | 2n-1 |
(2n-1)^{2} |
8x |
q |
+1 |
8q+1 |

1 | 1 | 1 | 8x | 0 | +1 | 1 |

2 | 3 | 9 | 8x | 1 | +1 | 9 |

3 | 5 | 25 | 8x | 3 | +1 | 25 |

4 | 7 | 49 | 8x | 6 | +1 | 49 |

5 | 9 | 81 | 8x | 10 | +1 | 81 |

6 | 11 | 121 | 8x | 15 | +1 | 121 |

7 | 13 | 169 | 8x | 21 | +1 | 169 |

8 | 15 | 225 | 8x | 28 | +1 | 225 |

9 | 17 | 289 | 8x | 36 | +1 | 289 |

10 | 19 | 361 | 8x | 45 | +1 | 361 |

We note that we can find a q value for each of the numbers, so that 8q+1 is equal to the square of the odd number.

It is possible that these q's look familiar in some way. After a short or long time, of thinking about this, we might realise that the q's are the sum of the natural numbers.

[4.6]

Substituting (2n-1)

[4.7]

Rearranging and expanding the square:

[4.8]

Adding similar terms and dividing by 8, we get

[4.9]

And adding n to both sides:

[4.10]

There is one issue, which we deal with in the next section: How can the following:

[4.4, repeated]

be the same, in some way, as the sum of the first natural numbers?

n | (n^{2}+n)/2 |
m | +/- | 2m^{2}+/-m |
---|---|---|---|---|

0 | 0 | 0 | 0 | |

1 | 1 | 1 | - | 1 |

2 | 3 | 1 | + | 3 |

3 | 6 | 2 | - | 6 |

4 | 10 | 2 | + | 10 |

5 | 15 | 3 | - | 15 |

Naturally, the minus signs alternate and the m values occur twice for each value. Apart from this, I see nothing except... it seems I need to relate the n values to the m values.

Of course, I do another table:

n | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|

m | 0 | 1 | 1 | 2 | 2 | 3 |

m-n | 0 | 0 | 1 | 1 | 2 | 2 |

n | 6 | 7 | 8 | 9 | 10 | 11 |

m | 3 | 4 | 4 | 5 | 5 | 6 |

m-n | 3 | 3 | 4 | 4 | 5 | 5 |

(The length of the table indicates that I did not see anything quickly).

However, the following relationship comes to mind: m=ceil(n/2)

[5.1]

Substituting this into 2m

[5.2]

If n is even, then ceil(n/2)=n/2, and the second term will be positive, giving our normal formula.

If n is odd,

[5.3]

Substituting this in 5.2, we get:

which simplifies to our normal equation for the sum of the natural numbers.

Of course, I prefer our old formula!

The sum of the first n odd natural numbers is (2k-1 represents any odd number):

[6.1]

We can expand the left-hand side:

[6.2]

And use our formula for the sum of the natural numbers:

[6.3]

Rounding up like terms, the sum of the first n odd natural numbers is:

[6.4]

Expanding the left-hand side:

[7.4]

Applying our formula for the sum of the first n natural numbers:

[7.5]

The sum of the first n even numbers is bigger than the sum of the first n odd numbers, because the first even number (2) is bigger than the first odd number (1) and this pattern continues (4 is bigger than 3). Clearly it is always bigger by n.

The sum of the odd number is bigger than the sum of the natural numbers, because, after 1, the odd numbers are bigger than the corresponding natural number (3 is bigger than 2, and 5 is bigger than 3).

[7.5]

The sum of part of the series of natural numbers from n

[7.6]

Substituting the formula for the first n natural numbers in 7.6, we get:

[7.7]

Which gives us:

[7.8]

Collecting like terms:

[7.9]

Factorising gives us the formula for the series of natural numbers from n

Ken Ward's Mathematics Pages

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