How to Calculate the Value of a Given Sum in Mathematics?

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In summary, the process for finding the value of a given sum involves identifying the numbers being added and performing the necessary mathematical operations. To check if the correct value has been found, one can plug the answer back into the original equation or use a calculator or second method. If unsure of the numbers or operations involved, breaking the problem down or consulting a teacher or online resources can help. Common mistakes to watch out for include forgetting to carry over numbers, using the wrong order of operations, and misreading numbers. Different methods, such as using a number line, mental math strategies, or a calculator, can be used to find the value of a given sum.
  • #1
hxthanh
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Put $1\le n\in\mathbb Z$
Find the Sum:
$S_n=\displaystyle \sum_{k=1}^n\dfrac{2k+1-n}{(k+1)^2(n-k)^2+1}$
 
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  • #2
Re: Find the Sum

My solution
Denote $j=n-k-1$ then $k=1 \to j=n-2 \quad ;k=n \to j=-1$.
We get:
\begin{array}{rcl}S_n &=& \sum\limits_{k = 1}^n {\frac{{2k + 1 -n}}{{{{\left( {n - k} \right)}^2}{{\left( {k + 1} \right)}^2} + 1}}} = \sum\limits_{j = - 1}^{n - 2} {\frac{{n - 1 - 2j}}{{{{\left( {j + 1} \right)}^2}{{\left( {n - j} \right)}^2} + 1}}} \\\Rightarrow 2S_n &=& \sum\limits_{k = 1}^n {\frac{{2k + 1 - n}}{{{{\left( {n - k} \right)}^2}{{\left( {k + 1} \right)}^2} + 1}}} + \sum\limits_{k = - 1}^{n - 2} {\frac{{n - 1 - 2k}}{{{{\left( {k + 1} \right)}^2}{{\left( {n - k} \right)}^2} + 1}}} \\&=& n + 1 + \frac{{n - 1}}{{{n^2} + 1}} + \sum\limits_{k = 1}^{n - 2} {\frac{{2k + 1 - n}}{{{{\left( {n - k} \right)}^2}{{\left( {k + 1} \right)}^2} + 1}}} \\&+& \sum\limits_{k = 1}^{n - 2} {\frac{{n - 1 - 2k}}{{{{\left( {n - k} \right)}^2}{{\left( {k + 1} \right)}^2} + 1}}} + \left( { n + 1} \right) + \frac{{n - 1}}{{{n^2} + 1}}\\\Rightarrow S_n &=& n +1 + \frac{{n - 1}}{{{n^2} + 1}}\\&=& \boxed{\displaystyle \frac{{ n\left( {{n^2} + n + 2} \right)}}{{{n^2} + 1}}}\end{array}
 

FAQ: How to Calculate the Value of a Given Sum in Mathematics?

What is the process for finding the value of a given sum?

The process for finding the value of a given sum involves identifying the numbers that are being added together and then performing the necessary mathematical operations to arrive at the final answer.

How do I know if I have found the correct value for a given sum?

You can check if you have found the correct value for a given sum by plugging the answer back into the original equation and seeing if it makes the equation true. You can also use a calculator or a second method to verify your answer.

What should I do if I am unsure of the numbers or operations involved in a given sum?

If you are unsure of the numbers or operations involved in a given sum, you can try breaking the problem down into smaller parts or using estimation to get an approximate answer. You can also consult a teacher or use online resources for help.

Are there any common mistakes to watch out for when finding the value of a given sum?

Some common mistakes to watch out for when finding the value of a given sum include forgetting to carry over numbers when adding, using the wrong order of operations, and misreading numbers. It is important to double check your work and be careful with your calculations.

Can I use different methods to find the value of a given sum?

Yes, there are multiple methods that can be used to find the value of a given sum, such as using a number line, using mental math strategies, or using a calculator. You can choose whichever method works best for you and the specific problem you are trying to solve.

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