Number of solutions to summation

  • Thread starter praharmitra
  • Start date
  • Tags
    Summation
In summary, the general formula for the number of solutions Tn to a1 + a2 + a3 + ... + an = k, where all variables are non-negative integers, can be expressed as a polynomial using the identities for sums. This can simplify the calculation and provide an easier formula.
  • #1
praharmitra
311
1
What is the general formula for a1 + a2 + a3 + ... + an = A, where all the variables a_i and A are non-negative integers.
 
Mathematics news on Phys.org
  • #2
There isn't an easier way to add numbers than to add them, unless the numbers are special in some way (eg, part of a arithmetic/geometric series)
 
  • #3
qntty said:
There isn't an easier way to add numbers than to add them, unless the numbers are special in some way (eg, part of a arithmetic/geometric series)

ahhhh... sorry, completely scrwed up the question. I meant,

What is the general formula for the Number Of Solutions Tn to a1 + a2 + a3 + ... + an = k, where all variables are non-negative integers?

:P, sorry for the mistake.

However, I have a solution, but I am convinced that i had read an easier formula somewhere.

This is my argument

1. a = k, with only one variable, has only one solution. So T1 = 1

2. a + b = k, has k + 1 solutions, as a can take any value from 0 to k, and the corresponding b will be fixed. T2 = k + 1

[tex]T_{2} = \sum_{k ' = 0} ^k 1[/tex]

3. a + b + c = k

a + b can take any value from 0 to k, and corresponding c will be fixed. Suppose a + b = k'. Then from the above argument no. 2, no. of solutions are k' + 1. Thus to get total no. of solutions of a+b+c = k, we must sum k' over all possible k's

[tex] T_{3} = \sum_{k' = 0} ^k (k' + 1)[/tex]

[tex] = T_{3} = \sum_{k'' = 0} ^k \sum_{k' = 0} ^{k''} 1[/tex]


4. for a+b+c+d = k, we can divide it into a+b+c = k', then into a+b = k'', and solve it similarly like above.


Thus in general we'll have n-1 such summations over variables k', k'', k''' etc..., i.e

[tex]T_{n} = \sum_{k^{n-1} = 0} ^k \sum_{k^{n-2} = 0} ^{k^{n-1}}... \sum_{k'' = 0} ^{k'''} \sum_{k' = 0} ^{k^{''}} 1[/tex]


However, I am sure I have read an easier formula somewhere. Could someone list that down?
 
  • #4
Last edited by a moderator:

1. What is a summation equation?

A summation equation is a mathematical expression that represents the addition of a series of numbers. It is denoted by the symbol Σ (sigma) and is commonly used to simplify and calculate the total of a large set of numbers.

2. How do you find the number of solutions to a summation equation?

The number of solutions to a summation equation can be found by evaluating the equation and solving for the variable. This can be done by using algebraic manipulation or by using a calculator or computer program.

3. Are there always multiple solutions to a summation equation?

No, there may not always be multiple solutions to a summation equation. The number of solutions depends on the values of the numbers being added and the complexity of the equation. In some cases, there may only be one unique solution.

4. Can a summation equation have an infinite number of solutions?

Yes, a summation equation can have an infinite number of solutions. This can occur when the summation involves an infinite series, such as the sum of all natural numbers or the sum of all positive integers.

5. How do the number of solutions to a summation equation affect its significance?

The number of solutions to a summation equation can indicate the complexity and significance of the equation. A large number of solutions may suggest a more complex and meaningful relationship between the numbers being added, while a smaller number of solutions may indicate a simpler equation with less significance.

Similar threads

  • General Math
Replies
6
Views
804
Replies
1
Views
909
  • General Math
Replies
2
Views
913
  • General Math
Replies
8
Views
772
  • General Math
Replies
7
Views
1K
Replies
3
Views
777
  • General Math
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
12
Views
948
  • Engineering and Comp Sci Homework Help
Replies
7
Views
818
Back
Top