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Chapter 7: Problem 67

Explain why $$ \sum_{m=1}^{999}\left(m^{5}-2 m+7\right)=\sum_{k=1}^{999}\left(k^{5}-2 k+7\right) . $$

Short Answer

Expert verified
The given equation is true because the variables \(m\) and \(k\) are simply placeholders representing indices within the summation, with identical bounds and operations applied. Since they only serve as a means to represent the indices and don't affect the overall summation, both expressions are equivalent. Hence, the equation \(\sum_{m=1}^{999}\left(m^{5}-2 m+7\right)=\sum_{k=1}^{999}\left(k^{5}-2 k+7\right)\) holds true.

Step by step solution

01

Understand the summation notation

The summation notation \(\sum\) is used to represent the sum of a series of terms. In this case, the sum of the terms of the function inside the brackets is taken from 1 to 999. The only difference between the two given expressions is the variables used within the summation (m and k).
02

Analyze the variables in the summation

In both cases, the variable (m or k) is only an index running from 1 to 999. In both expressions, the index is squared, multiplied by 2, and then added to 7, and all of those results are summed up. Note that the variable itself does not affect the result of the summation.
03

Compare the two expressions

Given that the variables m and k are merely placeholders for the indices in the summation, and their values do not carry any special significance, we can conclude that the two expressions are equivalent because they follow the same pattern and rules of summation. Therefore, we can safely say that: $$ \sum_{m=1}^{999}\left(m^{5}-2 m+7\right)=\sum_{k=1}^{999}\left(k^{5}-2 k+7\right) $$

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