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Chapter 4: Problem 23

Rewrite the sum using sigma notation. Do not evaluate. $$ 3+5+7+9+\cdots+23 $$

Short Answer

Expert verified
The sum of the consecutive odd numbers from 3 to 23 can be rewritten in sigma notation as \(\sum_{n=1}^{11} (2n + 1)\).

Step by step solution

01

Identify the pattern and general term of the sequence

The given sequence is formed by consecutive odd numbers starting with 3: \(3, 5, 7, 9, \ldots , 23\). To find the general term of this sequence, let's analyze the differences between its terms: - \(5 - 3 = 2\) - \(7 - 5 = 2\) - \(9 - 7 = 2\) Based on this pattern, we can conclude that the general term of the sequence can be written as \(a_n = 2n + 1\), where \(a_n\) represents the nth term of the sequence. Now, we need to determine the limits for the index.
02

Determine the limits of the index

To find the appropriate index limits, we should find the values of n for the first and last term of the sequence: - For the first term, set \(a_n = 3\): \(3 = 2n + 1\), so \(n = 1\). - For the last term, set \(a_n = 23\): \(23 = 2n + 1\), so \(n = 11\). Now that we have the general term and limits, we can write the sum in sigma notation.
03

Writing the sum in sigma notation

The given sum can be written in sigma notation using the general term and limits as follows: \[ \sum_{n = 1}^{11} (2n + 1) \] So the sum of this sequence of consecutive odd numbers from 3 to 23 is written in sigma notation as \(\sum_{n = 1}^{11} (2n + 1)\).

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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 list of numbers with a definite pattern where each term after the first is obtained by adding a constant, called the common difference, to the previous term. For example, in the sequence 2, 4, 6, 8, ..., the common difference is 2. The general form of an arithmetic sequence can be written as:
\( a_n = a_1 + (n - 1)d \),
where \( a_n \) is the nth term, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference. As we see in successive odd numbers, the common difference is 2. The sequence from the exercise, which starts with 3, truly is arithmetic because each subsequent odd number is exactly 2 more than the previous one.
Sum of a Series
The sum of an arithmetic series is the addition of all the terms in an arithmetic sequence. A formula for finding the sum of the first n terms, \( S_n \), of an arithmetic sequence is:
\( S_n = \frac{n}{2}(a_1 + a_n) \),
where \( a_1 \) is the first term, \( a_n \) is the nth term, and \( n \) is the number of terms. Alternatively, you can use the formula:
\( S_n = \frac{n}{2}[2a_1 + (n - 1)d] \),
which can be more practical when the last term is not readily known. This formula is beneficial when dealing with a sequence where only the first term and the common difference are known, much like with the provided sequence of odd numbers.
Consecutive Odd Numbers
Consecutive odd numbers form an arithmetic sequence where the common difference is always 2. For any odd number, the next one can be found by adding 2. In general, the nth odd number can be expressed as:
\( 2n - 1 \), starting with \( n = 1 \) for the number 1.
In this example, since our sequence starts with 3, the first term (when \( n = 1 \)) is \( 2(1) + 1 = 3 \), highlighting that we are dealing with an arithmetic sequence of odd numbers. The sequence ends with 23, and by using the formula for consecutive odd numbers, we can find the position of 23 in the sequence by solving \( 2n + 1 = 23 \), revealing 23 as the 11th term.

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Most popular questions from this chapter

According to data from the American Petroleum Institute, the U.S. Strategic Petroleum Reserves from the beginning of 1981 to the beginning of 1990 can be approximated by the function $$ S(t)=\frac{613.7 t^{2}+1449.1}{t^{2}+6.3} \quad 0 \leq t \leq 9$$ where \(S(t)\) is measured in millions of barrels and \(t\) in years, with \(t=0\) corresponding to the beginning of 1981 . Using the Trapezoidal Rule with \(n=9\), estimate the average petroleum reserves from the beginning of 1981 to the beginning of 1990 .

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Express the area of the region under the graph of the function f over the interval as the limit of a sum (use the right endpoints), (b) use a computer algebra system (CAS) to find the sum obtained in part (a) in compact form, and (c) evaluate the limit of the sum found in part (b) to obtain the exact area of the region. $$ f(x)=\sin x ; \quad\left[0, \frac{\pi}{2}\right] $$

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Velocity of an Attack Submarine The following data give the velocity of an attack submarine taken at 10 -min intervals during a submerged trial run. $$\begin{array}{|l|ccccccc|} \hline \text { Time } t \text { (hr) } & 0 & \frac{1}{6} & \frac{1}{3} & \frac{1}{2} & \frac{2}{3} & \frac{5}{6} & 1 \\ \hline \text { Velocity } v \text { (mph) } & 14.2 & 24.3 & 40.2 & 45.0 & 38.5 & 27.6 & 12.8 \\ \hline \end{array}$$ Use Simpson's Rule to estimate the distance traveled by the submarine during the 1 -hr submerged trial run.
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