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Chapter 5: Problem 11

Express the sums in Exercises \(11-16\) in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. $$1+2+3+4+5+6$$

Short Answer

Expert verified
The sum can be expressed as \(\sum_{n=1}^{6} n\).

Step by step solution

01

Identify the Sequence

Observe the sequence provided: \(1, 2, 3, 4, 5, 6\). Notice that this is an arithmetic sequence where each term increases by 1.
02

Determine the General Term

The general term for an arithmetic sequence can be expressed as \(a_n = a_1 + (n-1) \cdot d\), where \(a_1\) is the first term and \(d\) is the common difference. Here, \(a_1 = 1\) and \(d = 1\), so the general term is: \(a_n = 1 + (n-1)\cdot 1 = n\).
03

Choose Lower Limit and Write in Sigma Notation

The lower limit (starting value) for \(n\) is typically 1. The sequence given ends with 6, so the upper limit for \(n\) will be 6. Therefore, the sum can be represented in sigma notation as \(\sum_{n=1}^{6} n\).

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Key Concepts

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

Arithmetic Sequence
An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant is called the common difference. For instance, consider the sequence given in the exercise: 1, 2, 3, 4, 5, 6. Here, we can clearly see that the common difference is 1, obtained by subtracting any term from the one that follows it.

Key characteristics of arithmetic sequences include:
  • A starting value, known as the first term.
  • A common difference that is added to each term to get the next.
  • An explicit formula to find any term in the sequence.
Understanding the arithmetic sequence is essential, especially when finding sums of sequence terms, which is often accomplished using sigma notation.
General Term
The general term is a formula that allows us to find any term in the sequence without listing all preceding terms. This is especially useful when dealing with large sequences. To find the general term in an arithmetic sequence, you use the formula:\[ a_n = a_1 + (n-1) \cdot d \]where:
  • \(a_n\) = the nth term you want to find,
  • \(a_1\) = the first term of the sequence, and
  • \(d\) = the common difference.
Using the sequence from the exercise, which starts at 1 and increases by 1 each time, the first term \(a_1\) is 1 and \(d\) is also 1. Plugging these into the formula, we find:\[ a_n = 1 + (n-1) \cdot 1 = n \]This means that in our sequence, the nth term is simply \(n\), making it straightforward to express the entire sequence compactly.
Summation
Summation, often represented by the sigma (\(\Sigma\)) notation, is a way of adding together a series of numbers. In the case of sequences, it allows us to sum all terms from the starting point to the ending point efficiently.The sigma notation is structured as:\[ \Sigma_{i=m}^{n} a_i \]where:
  • \(m\) is the starting value of the index \(i\),
  • \(n\) is the ending value of the index, and
  • \(a_i\) represents the general term of the sequence.
For our example with the sequence 1, 2, 3, 4, 5, 6, the summation would look like:\[ \Sigma_{n=1}^{6} n \]This notation compactly represents the sum of all terms from 1 to 6. Summation is a powerful tool in mathematics, notably simplifying the representation and calculation of large sums.

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Most popular questions from this chapter

Solve the initial value problems in Exercises \(55-60\) $$ \frac{d s}{d t}=8 \sin ^{2}\left(t+\frac{\pi}{12}\right), \quad s(0)=8 $$

True, sometimes true, or never true? The area of the region between the graphs of the continuous functions \(y=f(x)\) and \(y=g(x)\) and the vertical lines \(x=a\) and \(x=b(a

Upper and lower sums for increasing functions $$ \begin{array}{l}{\text { a. Suppose the graph of a continuous function } f(x) \text { rises steadily }} \\ {\text { as } x \text { moves from left to right across an interval }[a, b] . \text { Let } P} \\ {\text { be a partition of }[a, b] \text { into } n \text { subintervals of equal length }} \\ {\Delta x=(b-a) / n . \text { Show by referring to the accompanying figure }}\\\ {\text { that the difference between the upper and lower sums for }} \\ {f \text { on this partition can be represented graphically as the area }} \\\ {\text { of a rectangle } R \text { whose dimensions are }[f(b)-f(a)] \text { by } \Delta x \text { . }} \\ {\text { (Hint: The difference } U-L \text { is the sum of areas of rectangles }}\\\\{\text { whose diagonals } Q_{0} Q_{1}, Q_{1} Q_{2}, \ldots, Q_{n-1} Q_{n} \text { lie approximately }} \\ {\text { along the curve. There is no overlapping when these rectan- }} \\ {\text { gles are shifted horizontally onto } R . \text { . }}\end{array} $$ $$ \begin{array}{l}{\text { b. Suppose that instead of being equal, the lengths } \Delta x_{k} \text { of the }} \\ {\text { subintervals of the partition of }[a, b] \text { vary in size. Show that }}\end{array} $$ $$ U-L \leq|f(b)-f(a)| \Delta x_{\max } $$ $$ \begin{array}{l}{\text { where } \Delta x_{\max } \text { is the norm of } P, \text { and hence that } \lim _{ \| P | \rightarrow 0}} \\\ {(U-L)=0}\end{array} $$

Show that if \(f\) is continuous, then \(\int_{0}^{1} f(x) d x=\int_{0}^{1} f(1-x) d x\)

Solve the initial value problems in Exercises \(55-60\) $$ \frac{d s}{d t}=12 t\left(3 t^{2}-1\right)^{3}, \quad s(1)=3 $$
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