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Chapter 13: Problem 62

Seats in an Amphitheater An outdoor amphitheater has 35 seats in the first row, 37 in the second row, 39 in the third row, and so on. There are 27 rows altogether. How many can the amphitheater seat?

Short Answer

Expert verified
The amphitheater can seat 1647 people.

Step by step solution

01

- Identify the Sequence Type

Recognize that the number of seats in each row forms an arithmetic sequence because the number increases by a constant difference. The first term, denoted as \(a_1\), is 35. The common difference, \(d\), is 2.
02

- Determine the Number of Rows

The problem states there are 27 rows in total. This means \(n = 27\).
03

- Find the Number of Seats in the Last Row

Use the formula for the nth term of an arithmetic sequence: \(a_n = a_1 + (n - 1) d\). Substituting the values: \[a_{27} = 35 + (27 - 1) \cdot 2 = 35 + 52 = 87\]. So, the 27th row has 87 seats.
04

- Use the Formula for Sum of an Arithmetic Series

The sum of the first \(n\) terms of an arithmetic sequence is given by: \(S_n = \frac{n}{2} (a_1 + a_n)\). Substituting the values: \[S_{27} = \frac{27}{2} (35 + 87)\]
05

- Calculate the Total Number of Seats

Perform the calculations: \[S_{27} = \frac{27}{2} \cdot 122 = 27 \cdot 61 = 1647\]. Therefore, the total number of seats in the amphitheater is 1647.

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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 series of numbers where the difference between consecutive terms is always constant. This difference is known as the common difference and is typically denoted by the letter \(d\). For instance, in the provided exercise, the number of seats in each row of the amphitheater increases by 2, which is the common difference. The sequence starts with 35 seats (the first term denoted by \(a_1\)) and continues by adding the common difference to each term to get the next one.
So the sequence looks like this:
\[35, 37, 39, 41, \ldots\]
This regular pattern makes it easier to predict future terms or analyze data.
sum of arithmetic series
To find the total number of seats in the amphitheater, we need to calculate the sum of the arithmetic series.
The sum of an arithmetic series is the total when all terms in the sequence are added together. For calculation, we can use the formula:
\[S_n = \frac{n}{2}(a_1 + a_n)\]
where \(S_n\) represents the sum of the first \(n\) terms, \(a_1\) is the first term, and \(a_n\) is the nth term.
Using the formula, we find the last term of the sequence using:
\[a_n = a_1 + (n - 1) d\]
In this exercise, we have 27 rows (n = 27), and thus:\[a_1 = 35\] (first term) \(d = 2\) (common difference), resulting in the nth term:
\[a_{27} = 35 + (27 - 1) \cdot 2 = 87\]
Summing up, we substitute the values into the sum formula:
\[S_{27} =\frac{27}{2}\times(35 + 87)= 1647\]
Therefore, the total number of seats is 1647.
sequence and series
Before diving into solving problems, it's helpful to distinguish between a sequence and a series.
A sequence is an ordered list of numbers where each number is called a term. It can be finite or infinite. For example, the number of seats per row in the amphitheater forms a finite arithmetic sequence:
\[35, 37, 39, 41, \ldots, 87\]
On the other hand, a series is the sum of the terms in a sequence. When we add up all the terms in the sequence of seats, it becomes an arithmetic series. So, the exercise technically deals with an arithmetic series because it ultimately asks for the sum of all the rows of seats.
This distinction is crucial as it guides which formulas and methods to use for finding solutions.

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Most popular questions from this chapter

Bode's Law In \(1772,\) Johann Bode published the following formula for predicting the mean distances, in astronomical units (AU), of the planets from the sun: $$ a_{1}=0.4 \quad a_{n}=0.4+0.3 \cdot 2^{n-2} $$ where \(n \geq 2\) is the number of the planet from the sun. (a) Determine the first eight terms of the sequence. (b) At the time of Bode's publication, the known planets were Mercury \((0.39 \mathrm{AU}),\) Venus \((0.72 \mathrm{AU}),\) Earth \((1 \mathrm{AU})\) Mars \((1.52 \mathrm{AU}),\) Jupiter \((5.20 \mathrm{AU}),\) and Saturn \((9.54 \mathrm{AU})\) How do the actual distances compare to the terms of the sequence? (c) The planet Uranus was discovered in \(1781,\) and the asteroid Ceres was discovered in \(1801 .\) The mean orbital distances from the sun to Uranus and Ceres " are \(19.2 \mathrm{AU}\) and \(2.77 \mathrm{AU},\) respectively. How well do these values fit within the sequence? (d) Determine the ninth and tenth terms of Bode's sequence. (e) The planets Neptune and Pluto" were discovered in 1846 and \(1930,\) respectively. Their mean orbital distances from the sun are \(30.07 \mathrm{AU}\) and \(39.44 \mathrm{AU},\) respectively. How do these actual distances compare to the terms of the sequence? (f) On July \(29,2005,\) NASA announced the discovery of a dwarf planet \((n=11),\) which has been named Eris. Use Bode's Law to predict the mean orbital distance of Eris from the sun. Its actual mean distance is not yet known, but Eris is currently about 97 astronomical units from the sun.

Koch's Snowflake The area inside the fractal known as the Koch snowflake can be described as the sum of the areas of infinitely many equilateral triangles, as pictured below. For all but the center (largest) triangle, a triangle in the Koch snowflake is \(\frac{1}{9}\) the area of the next largest triangle in the fractal. Suppose the area of the largest triangle has area of 2 square meters. (a) Show that the area of the Koch snowflake is given by the series $$A=2+2 \cdot 3\left(\frac{1}{9}\right)+2 \cdot 12\left(\frac{1}{9}\right)^{2}+2 \cdot 48\left(\frac{1}{9}\right)^{3}+2 \cdot 192\left(\frac{1}{9}\right)^{4}+\cdots$$ (b) Find the exact area of the Koch snowflake by finding the sum of the series.

Find the first term and the common difference of the arithmetic sequence described. Find a recursive formula for the sequence. Find a formula for the nth term. 8th term is 4; 18th term is - 96

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms. $$ \left\\{\left(\frac{5}{4}\right)^{n}\right\\} $$

Find the first term and the common difference of the arithmetic sequence described. Find a recursive formula for the sequence. Find a formula for the nth term. 14th term is \(-1 ; 18\) th term is -9
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