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Chapter 10: Problem 16

Find the terms \(a_{0}, a_{1},\) and \(a_{2}\) for each sequence. $$a_{n}=6\left(5^{n}\right)$$

Short Answer

Expert verified
The terms \(a_{0}, a_{1},\) and \(a_{2}\) of the sequence are 6, 30 and 150 respectively.

Step by step solution

01

Find the term \(a_{0}\)

To find the term \(a_{0}\), substitute \(n = 0\) into the formula \(a_{n}=6\left(5^{n}\right)\). So, \(a_{0} = 6 \cdot (5^{0}) = 6 \cdot 1= 6\).
02

Find the term \(a_{1}\)

To find the term \(a_{1}\), substitute \(n = 1\) into the formula \(a_{n}=6\left(5^{n}\right)\). So, \(a_{1} = 6 \cdot (5^{1}) = 6 \cdot 5 = 30\).
03

Find the term \(a_{2}\)

To find the term \(a_{2}\), substitute \(n = 2\) into the formula \(a_{n}=6\left(5^{n}\right)\). So, \(a_{2} = 6 \cdot (5^{2}) = 6 \cdot 25 = 150\).

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Key Concepts

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

Sequence Terms
Understanding sequence terms is crucial for analyzing and interpreting geometric sequences. In mathematics, a sequence is an ordered list of numbers, where each number is called a term. Specifically, a geometric sequence occurs when each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
For the exercise above, the sequence is defined by the formula \(a_{n} = 6 \times 5^n\). This means:
  • \(a_{0}\) - The zero term in the sequence, calculated by setting \(n = 0\).
  • \(a_{1}\) - The first term in the sequence by setting \(n = 1\).
  • \(a_{2}\) - The second term by setting \(n = 2\).
Each term is derived using the position number \(n\) and the sequence formula, which allows us to systematically find any other terms in the sequence.
Exponential Functions
Exponential functions play an essential role in many mathematical sequences due to their ability to model growth processes, such as in geometric sequences. An exponential function typically takes the form \(f(x) = ab^x\), where \(a\) is a constant, \(b\) is the base of the exponential, and \(x\) is the exponent or variable.
In the exercise's geometric sequence formula \(a_{n} = 6 \times 5^n\):
  • \(6\) is the constant that scales the sequence in size and position, known here as the initial value since \(n = 0\) gives \(a_0 = 6\).
  • \(5\) is the base and acts as the common ratio, revealing how quickly the sequence grows.
The term calculated with \(n\) as the exponent exponentially increases by multiplying the previous term by \(5\). An understanding of exponential functions is key for predicting how sequences expand, model patterns, and solve real-world problems.
Recursive Formulas
Recursive formulas provide an alternative way to represent sequences by calculating each term based on previous ones, rather than with a standalone expression like explicit formulas. A recursive formula for geometric sequences might look different from the explicit format:
  • For instance, starting with \(a_0 = 6\), and then providing the recursive relation \(a_{n} = a_{n-1} \times 5\).
  • This indicates that each term can be found by multiplying the term before it by the common ratio \(5\), after establishing the initial term \(a_0 = 6\).
Recursive formulas require you to know at least one previous term before proceeding to calculate the next, making memorization of one term and the multiplication step essential. While slightly different from explicit formulas, recursive methods foster a deeper understanding of sequence behavior over successive terms, emphasizing the steady scale transformation brought about by the common ratio.

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Most popular questions from this chapter

In this set of exercises, you will use sequences to study real-world problems. Sports The men's and women's U.S. Open tennis tournaments are elimination tournaments. Each tournament starts with 128 players in 64 separate matches. After the first round of competition, 64 players are left. The process continues until the final championship match has been played. (a) What type of sequence gives the number of players left after each round? (b) How many rounds of competition are there in each tournament?

In a telephone survey, people are asked whether they have seen each of four different films. Their answers for each film (yes or no) are recorded. (a) What is the sample space? (b) What is the probability that a respondent has seen exactly two of the four films? (c) Assuming that all outcomes are equally likely, what is the probability that a respondent has seen all four films?

The Woosamotta University bookstore sells "W. U." T-shirts in four sizes: \(\mathrm{S}, \mathrm{M},\mathrm{L},\) and \(\mathrm{XL}\). Both the blue and yellow shirts are available in all four sizes, but the red shirts come in only small and medium. What is the minimum number of W. U. T-shirts the bookstore should stock if it wishes to have available at least one of each size and color?

In this set of exercises, you will use sequences and their sums to study real- world problems. A carpet warehouse needs to calculate the diameter of a rolled carpet given its length, width, and thickness. If the diameter of the carpet roll can be predicted ahead of time, the warehouse will know how much to order so as not to exceed warehouse capacity. Assume that the carpet is rolled lengthwise. The crosssection of the carpet roll is then a spiral. To simplify the problem, approximate the spiral cross-section by a set of \(n\) concentric circles whose radii differ by the thickness \(t\) Calculate the number of circles \(n\) using the fact that the sum of the circumferences of the \(n\) circles must equal the given length. How can you find the diameter once you know \(n ?\)

How many different six-letter arrangements are there of the letters in the word PIPPIN? This exercise involves a slightly different strategy than the strategies discussed in the Examples. First draw six slots for six letters. In how many ways can you put the three P's in the slots? You have three slots left over. In how many ways can you place the two I's? The last slot, by default, will contain the N .
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