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Chapter 5: Problem 44

Write a formula for the general term (the nth term) of each arithmetic sequence. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 the term of the sequence. \(6,1,-4,-9, \ldots\)

Short Answer

Expert verified
The 20th term of the arithmetic sequence is -89.

Step by step solution

01

Identify the common difference

In an arithmetic sequence, every term is obtained from the previous one by adding a constant difference. To identify this, subtract the second term from the first term, then the third term from the second, and so on. Here, 1-6 is equal to -5, and -4-1 is also equal to -5. Since the differences are the same, -5 is the common difference (d).
02

Formulate the general term

The general term, \(a_{n}\), of an arithmetic sequence can be written as \(a_{n}= a_{1}+ (n-1) * d\), where \(a_{1}\) is the first term and d is the common difference. Here, the first term, \(a_{1}\) = 6, and the common difference, d = -5. Substituting these values in, the formula becomes \(a_{n} = 6 - 5(n - 1)\).
03

Find the 20th term

Now that we have the general formula, we can find the 20th term by substituting \(n = 20\) into the formula: \(a_{20} = 6 - 5(20 - 1) = -89\). Therefore, the 20th term, \(a_{20}\), of the arithmetic sequence is -89.

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Key Concepts

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

Common Difference in Arithmetic Sequences
Understanding the concept of common difference is crucial when dealing with arithmetic sequences. It's defined as the constant amount that each term in the sequence changes by from one to the next. In the arithmetic sequence provided, to find this value, one typically subtracts a term from the term that follows it. For instance, if our sequence is given as \(6, 1, -4, -9, \ldots\), we calculate the common difference (\(d\)) by subtracting the second term from the first ((\(1-6 = -5\))), and to confirm, subtracting the third term from the second ((\(-4-1 = -5\))).\
\
Repeatedly finding the same result of \( -5 \) confirms the consistency of the common difference across terms. Notably, if the difference wasn't constant, it wouldn't be an arithmetic sequence. Knowing \(d\) is the pillar for understanding the sequence's behavior, as it allows prediction of future terms and even reverse engineering of past terms.
General Term Formula of an Arithmetic Sequence
The general term formula is vital for finding any term in an arithmetic sequence without having to write out the entire sequence. This formula is expressed as \( a_n = a_1 + (n - 1)d \), where \( a_n \) is the term you want to find, \( a_1 \) is the first term of the sequence, \( n \) is the term's position in the sequence, and \( d \) is the common difference.

When constructing the formula for the provided sequence \(6, 1, -4, -9, \ldots\), we already determined the first term \( a_1 \) as 6 and the common difference \( d \) as -5. Plugging these into the formula gives us \( a_n = 6 + (n - 1)(-5) \). This formula is the blueprint for calculating any term's value in the sequence efficiently.
Sequence Term Calculation
The real power of having a general term formula is seen when calculating specific terms in a sequence, like the 20th term (\(a_{20}\)). This is done by simply substituting the desired term position into the formula. For our sequence, using \( a_n = 6 + (n - 1)(-5) \) and substituting \( n \) with 20 gives us \( a_{20} = 6 + (20 - 1)(-5) \).

Carrying out the arithmetic, \( a_{20} = 6 - 95 \), which simplifies to \( a_{20} = -89 \). Hence, \(a_{20}\), or the 20th term of the sequence, is -89. This demonstrates the ease of finding distant terms in the sequence without computing each preceding term, showing the utility and convenience of the arithmetic sequence's general term formula.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If a sequence is geometric, we can write as many terms as we want by repeatedly multiplying by the common ratio.

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=20, r=-4\)

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7}\), the seventh term of the sequence. \(12,6,3, \frac{3}{2}, \ldots\)

In Exercises 91-98, write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7}\), the seventh term of the sequence. \(3,12,48,192, \ldots\)

In Exercises 133-134, you will develop geometric sequences that model the population growth for California and Texas, the two most populated U.S. states. The table shows the population of California for 2000 and 2010 , with estimates given by the U.S. Census Bureau for 2001 through \(2009 .\) $$ \begin{aligned} &\begin{array}{|l|c|c|c|c|c|} \hline \text { Year } & \mathbf{2 0 0 0} & \mathbf{2 0 0 1} & \mathbf{2 0 0 2} & \mathbf{2 0 0 3} & \mathbf{2 0 0 4} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 33.87 & 34.21 & 34.55 & 34.90 & 35.25 \\ \hline \end{array}\\\ &\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Year } & \mathbf{2 0 0 5} & \mathbf{2 0 0 6} & \mathbf{2 0 0 7} & \mathbf{2 0 0 8} & \mathbf{2 0 0 9} & \mathbf{2 0 1 0} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 35.60 & 36.00 & 36.36 & 36.72 & 37.09 & 37.25 \\ \hline \end{array} \end{aligned} $$ a. Divide the population for each year by the population in the preceding year. Round to two decimal places and show that California has a population increase that is approximately geometric. b. Write the general term of the geometric sequence modeling California's population, in millions, \(n\) years after \(1999 .\) c. Use your model from part (b) to project California's population, in millions, for the year 2020 . Round to two decimal places.
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