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Chapter 12: Problem 76

Explain what each part of the notation \(\sum_{k=1}^{12} 2 k\) means.

Short Answer

Expert verified
The Sigma symbol signifies summation, \(k\) is the index of summation from 1 to 12, and \(2k\) is the expression being summed.

Step by step solution

01

Understanding the Sigma Notation

The symbol \(\textbackslash sum\) is called Sigma notation and it signifies the summation of a sequence of numbers.
02

Identifying the Index of Summation

The variable \(k\) below the Sigma symbol is the index of summation, which indicates the starting value of the sequence, in this case, \(k=1\).
03

Setting the Upper Limit of Summation

The number 12 on top of the Sigma symbol shows the upper limit of the summation, meaning \(k\) will take on values from 1 to 12.
04

Analyzing the Expression to be Summed

The expression \(2k\) to the right of the Sigma symbol is the term that will be summed as \(k\) ranges from 1 to 12.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Summation
Summation is a mathematical operation that adds up a series of numbers. The symbol for summation is the Greek letter Sigma (\textbackslash sum). It tells us to add together a sequence of terms. For example, in \(\textbackslash sum\_ \{k=1\}\^ \{12\} 2k\), the Sigma notation signifies that we are summing the expression \(\textbackslash 2k \) over a specific range, from k = 1 to k = 12. This means we calculate the values of 2k for each k from 1 up to 12 and then add them all together. Summation is widely used in various mathematical computations and is a fundamental concept in calculus and higher-level mathematics.
Index of Summation
The index of summation is the variable that changes as we progress through the sequence we are summing. It is typically denoted by a letter like k, i, or j. In our example, \(\textbackslash sum\_ \{k=1\}\^ \{12\} 2k \), the index of summation is k. It starts at the lower limit, 1, and increases by 1 until it reaches the upper limit, 12. The index helps us know which values to plug into the expression being summed, \(\textbackslash 2k\). By systematically changing the index, we can evaluate the sequence term by term.
Upper Limit of Summation
The upper limit of summation is the value at which the summation stops. It tells us the final number that the index of summation will reach. In our example, \(\textbackslash sum\_ \{k=1\}\^ \{12\} 2k \), the upper limit is 12. This means k will take on values from 1 to 12, inclusive. The upper limit is written above the Sigma symbol. It is crucial because it defines the total number of terms to be summed. With an upper limit of 12, we sum the terms \(\textbackslash 2\times 1\), \(\textbackslash 2\times 2\), ..., up to \(\textbackslash 2\times 12\).
Sequence of Numbers
A sequence of numbers is an ordered list of numbers that often follows a specific pattern or rule. In the context of summation, a sequence is generated by evaluating an expression for different values of the index of summation. For the example, \(\textbackslash sum\_ \{k=1\}\^ \{12\} 2k \), we evaluate the expression \(\textbackslash 2k \) for k=1, k=2, up to k=12. This gives us the sequence \([2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]\). By summing these numbers, we capture the total accumulated value of the sequence, which showcases the power and utility of summation in mathematics.

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