Amounts - Closed Form - Where to Start
I am trying to understand the basics as it has to do with forming a closed form expression from a sum. I understand the goal, but I do not understand what to follow in order to achieve the goal.
Find a closed form for the sum k + 2k + 3k + ... + K ^ 2. Prove your claim
My first approach was to turn it into a recurrent relationship that doesn't cleanly work. After that, I would try to move from a recurrent to a closed form, but I was unsuccessful there.
Does anyone know of a strong approach to solving problems like this? Or any simplified tutorials that might be provided? The material I find on the internet does not help and causes further confusion.
thank
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Nobody gave a mathematical approach, so I am adding a mathematical approach to this AP problem.
This series is equal to 1k + 2k + 3k + .... + kk (OR k ^ 2)
Therefore, this means that this series contains only k terms.
Further, since here all consecutive terms are larger than the previous term due to the constant total difference, i.e. ... k
So this is an arithmetic progression.
Now, to calculate the total sum, the formula is defined by the formula: -
S(n) = n/2{a(1)+a(n)}
where S (n) is the summation of series up to n terms
n is the number of members in the series,
a (1) is the first term in the series, and a (n) is the last (n th ) term in the series.
Here, substituting the terms of this series in the summation formula, we get: -
S (n) = k / 2 {1k + kk} = (k / 2) {k + k ^ 2) = * . [(k^2)/2 + (k^3)/2]
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Assad explained the mathematical approach in the comments to address this question.
If you are interested in a programming approach that works for more complex expressions, you can use Sympy in Python.
For example:
import sympy
x,k = sympy.symbols('x k')
print sympy.sum(x*k,(x,1,k))
prints:
k*(k/2 + k**2/2)
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If you are interested in the general algorithm for calculating such sums (and more complex ones), I cannot recommend the book A = B. is enough.
The authors were kind enough to make the PDF freely available:
http://www.math.upenn.edu/~wilf/AeqB.html
Enjoy!
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