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User:Email4mobile/sum of power arithmetic progression

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I'm trying to call this "sum of power arithmetic progression" because it is a generalized arithmetic progression for powers. (i.e.: 1k+2k+3k+...).

One can generalize this in the following form:

Since we are discussing a sequence of natural numbers, then we expect that the total sum is on the form of a polynomial.

or simply


Using the previous relation, one can find the values of the terms: a1, a2,...,an by verifying a minimum of k+1 different values for the formula (example n=1, 2, 3, ..., k+1) and then putting them into k+1 linear equations to be solved.

Examples

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The simplest form of this form is the normal arithmetic progression, k = 1.

This is well known, where a1 = a2 = 0.5

To prove this, assume the solution for n= 1, 2.

For n = 1,

or

With n = 2

or

Solving both equations results:a1 = a2 = 0.5

Again, we can solve the following power series for k = 2

For n = 1,

or

...(1)

With n = 2

or

...(2)

With n = 3

or

...(3)

Solving the 3 equations results:a1 = 1/6,a2 = 1/2,a3 = 1/3

List of some sum terms

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k Terms Polynomial form
1 [1/2, 1/2]
2 [1/6, 1/2, 1/3]
3 [0, 1/4, 1/2, 1/4]
4 [-1/30, 0, 1/3, 1/2, 1/5]
5 [0, -1/12, 0, 5/12, 1/2, 1/6]
6 [1/42, 0, -1/6, 0, 1/2, 1/2, 1/7]
7 [0, 1/12, 0, -7/24, 0, 7/12, 1/2, 1/8]
8 [-1/30, 0, 2/9, 0, -7/15, 0, 2/3, 1/2, 1/9]
9 [0, -3/20, 0, 1/2, 0, -7/10, 0, 3/4, 1/2, 1/10]

Notes

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Looking carefully at the table, checking terms, we can realize that:

is always 1/(k+1),

is always 1/2,

is always positive,

is always 0,

is always negative,

is always 0,

is always positive,

... and so on!

Script

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Note: I generated the previous table values using my online Math wizard (JScript based). The script to be used in the command line is:

k=1;  // k can take 1, 2, 3, ...
m=[];
m[k+1]=1;
for(i=2;i<=k+1;i++)
 m[i*(k+2)-1] = pow(i,k)+m[(i-1)*(k+2)-1];
  for(i=0;i<=k;i++)
   for(j=i*(k+2);j<=k+i*(k+2);j++)m[j]=pow(i+1,j-i*(k+2)+1);
    msolve(matrix(k+1,k+2,m))