# Symmetrical Summation with Central Point | Solving for a(0) to a(N-1/2)

• JeeebeZ
In summary: It seems to me that the conversation is about finding a function that satisfies certain properties, rather than trying to solve a specific problem. In summary, the conversation is about finding a function that satisfies certain properties, specifically taking the values 1 and 2 and being symmetrical from the center point. The suggested solution is to use a normalized sinc function, but it is questioned whether a continuous function is necessary for this scenario.
JeeebeZ

## Homework Statement

I need a summation where the answer is 1 2 2 2 2 2 2 2

## Homework Equations

a(0) + sum(2*a(1) + 2*a(2) +2*a(3))

## The Attempt at a Solution

I unfortunately have no idea where to start, basically it is taking a symmetrical function from 0 to N-1. where the function is symmetric from the center point.

I have an answer for it which is

a(N-1/2) + sum_n=1^(N-1)/2 (2*a[((N-1)/2) - n])

However I wanted to put it all inside the summation

I've no idea what you mean.
In what sense does a summation have an answer like "1 2 2 2 2 2 2 2" rather than a single number?
What does a(0) + sum(2*a(1) + 2*a(2) +2*a(3)) mean? Do you mean a(0) + sum(2*a(1) + 2*a(2) +2*a(3) + ...)? If so, how is that different from 2(∑an) - a0?
From the mention of symmetry, I'm guessing this comes from ##\Sigma_{n=-N}^Na_n##, where a-n = an, but I still don't understand what you wish to achieve.
What quantity does your final equation represent?

Sorry, the "1 2 2 2 2 ..." bit was basically was how I wanted each part of the summation to equate to, so essentially it would be "1 + 2 + 2 + 2 ..."

"2(∑an) - a0" is the exact same as "a0 + 2(∑an)" with the exception that n goes from 0->N for the first one and 1->N for the second one.

Basically I want to achieve this, "2(∑an) - a0", without having to subtract the a0 at the end.

Still unclear.. are you looking for an algebraic function f(n) that takes the values f(0) = 1, f(n) = 2 for n > 1? So that you can write ∑n>=0anf(n)?

yes that's exactly what I'm looking to do

You want ##f(n) = 2-\textrm{sinc}(n)##, where sinc is the normalized sinc function:$$\textrm{sinc} (x) = \begin{cases} \frac {\sin(\pi x)}{\pi x}&x \ne 0\\ 1 & x = 0 \end{cases}$$

LCKurtz said:
You want ##f(n) = 2-\textrm{sinc}(n)##, where sinc is the normalized sinc function:$$\textrm{sinc} (x) = \begin{cases} \frac {\sin(\pi x)}{\pi x}&x \ne 0\\ 1 & x = 0 \end{cases}$$

Cute, but I don't see the value in this. As you note, that function has to have a separate definition at x=0. So why not use Kronecker delta? I see nothing in the thread to suggest a continuous function is required.
JeeebeZ, why do you want such a function? What ultimately are you trying to achieve?

haruspex said:
Cute, but I don't see the value in this.

Agreed. But then, I don't see any point to the original question either.

## 1. What is symmetrical summation with central point?

Symmetrical summation with central point is a mathematical technique used to calculate the sum of a sequence of numbers, where the central point is the middle term. This technique involves adding the first and last terms, then the second and second to last terms, and so on until the central term is reached. The sum is then multiplied by 2 to get the total sum of the sequence.

## 2. How is symmetrical summation with central point used in solving for a(0) to a(N-1/2)?

In order to solve for a(0) to a(N-1/2), symmetrical summation with central point is used to find the sum of the sequence. The formula for this is a(0) + a(1) + ... + a(N-1/2) = 2S, where S is the sum of the sequence. This allows for the individual values of a(0) to a(N-1/2) to be determined.

## 3. What is the importance of finding a(0) to a(N-1/2) in symmetrical summation with central point?

Finding a(0) to a(N-1/2) is important in symmetrical summation with central point because it allows for the complete solution of the sequence to be determined. These values represent the individual terms in the sequence and can be used in further calculations or analyses.

## 4. Can symmetrical summation with central point be used for non-symmetrical sequences?

No, symmetrical summation with central point is specifically designed for symmetrical sequences where the central term is the middle term. For non-symmetrical sequences, other mathematical techniques must be used to find the sum.

## 5. What are some real-world applications of symmetrical summation with central point?

Symmetrical summation with central point can be applied in various fields such as statistics, physics, and engineering. It can be used to calculate the center of mass, find the equilibrium point in a system, or determine the average value of a continuous data set. In statistics, it can be used to calculate the mean of a set of data. In physics, it can be used to find the total energy of a system.

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