Reduce the sequenceor how to calculate it efficiently

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In summary, the conversation discusses a mathematical sequence in summation notation and provides a simplified form for calculating its value for large values of n. The solution involves using a prime number modulus and suggests using Wolfram Alpha for evaluation if Mathematica is not available.
  • #1
kuldeepfouzda
1
0
Sequence is
(in Summation notation)
SUMMATION[from k=0 to k=floor(n/2)] (n-k)C(k) * 2(n-k-1)
refer to this image
http://i.snag.gy/tFI7e.jpg
after expanding it becomes
(n)C(0)*2(n-1) + (n-1)C(1)*2(n-2) + (n-2)C(2) * 2(n-3) +...
I want to calculate the value of function for a large value of n(up to
1010, and ans is to be found MODULUS to some prime number..like
1010+7).

Please simplify the series if possible or let me know the way how to
solve it efficiently.
 
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  • #2
Mathematica code

Code: 
Sum[Binomial[n - k, k]*2^(n - k - 1), {k, 0, Floor[n/2]}]

you can use wolfram alpha to evaluate the exact form, if you don't have mathematica installed.

answer is attached
 

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1. What is a sequence reduction and why is it important in scientific research?

A sequence reduction is a process used to simplify a series of data points or events in order to identify patterns or trends. It is important in scientific research as it allows for more efficient and accurate analysis of data, which can lead to new discoveries and advancements in various fields.

2. How can I calculate a sequence reduction efficiently?

The most efficient way to calculate a sequence reduction is by using algorithms specifically designed for this purpose. These algorithms take into account the size of the sequence and the type of data being analyzed, and can significantly reduce the time and effort required for calculation.

3. What factors should be considered when selecting an appropriate method for sequence reduction?

When selecting a method for sequence reduction, factors such as the type of data, the size of the sequence, and the desired level of accuracy should be taken into consideration. It is also important to consider the computational resources available and the complexity of the algorithm.

4. How can I ensure the accuracy of a sequence reduction?

To ensure the accuracy of a sequence reduction, it is important to use reliable and well-tested algorithms, and to validate the results using multiple methods. It is also recommended to use a large and diverse dataset for analysis and to carefully analyze any potential sources of error.

5. Are there any limitations to using sequence reduction in scientific research?

While sequence reduction can be a powerful tool in scientific research, it is important to be aware of its limitations. These include the potential loss of information and the need to carefully select appropriate algorithms for different types of data. It is also important to consider the potential biases that may arise from using a reduction method.

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