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Chapter 5: Problem 48

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 th term of the sequence. \(a_{1}=-70, d=-5\)

Short Answer

Expert verified
The general term of the given arithmetic sequence is \(a_{n} = -70 - 5 \times (n - 1)\) and its 20th term is \(a_{20} = -165\).

Step by step solution

01

Write down the formula for the general term of an arithmetic sequence

The formula for the general term of an arithmetic sequence is \(a_{n} = a_{1} + (n - 1) \times d\).
02

Substitute the given values into the general term formula

Substitute \(a_{1} = -70\), \(d = -5\) into the general term formula. This gives: \(a_{n} = -70 + (n - 1) \times -5\).
03

Simplify the general term formula

On simplifying the formula from step 2 we get: \(a_{n} = -70 - 5 \times (n - 1)\). This is our general term formula.
04

Calculate the 20th term

Substitute \(n = 20\) in the formula: \(a_{20} = -70 - 5 \times (20 - 1)\). Simplify this to find the 20th term: \(a_{20} = -70 - 5 \times 19 = -165\).

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Key Concepts

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

General Term Formula
When dealing with arithmetic sequences, a key concept is the general term formula. This formula allows us to find any term in the sequence. In an arithmetic sequence, each term after the first is generated by adding a constant value, known as the common difference, to the previous term.
The general term formula, represented as \(a_{n} = a_{1} + (n - 1) \times d\), is used to determine the nth term. This formula is powerful because it gives you the ability to find any term in the sequence without listing all preceding terms. To use this formula:
  • Identify \(a_1\), the first term in the sequence.
  • Determine \(d\), which is the common difference between terms.
  • Choose an \(n\), the term number you wish to find.
This approach simplifies the process, especially when dealing with large indices, saving time and effort.
Common Difference
The term 'common difference' refers to the constant value added to each term of an arithmetic sequence to get to the next term. It is a fundamental aspect of these sequences, as it determines the uniform rate at which each term progresses.
To find the common difference, simply subtract any term from the succeeding term in the sequence. For example, in the given problem, the common difference \(d\) is \(-5\). This means each term decreases by 5 from the previous term.
Understanding the common difference is crucial because it's used in the general term formula, \(a_{n} = a_{1} + (n - 1) \times d\).
  • A positive \(d\) indicates an increasing sequence.
  • A negative \(d\) signifies a decreasing sequence.
  • A zero \(d\) means all terms are identical.
Knowing how to determine and use the common difference helps in accurately predicting the components of the sequence.
Nth Term Calculation
Calculating the nth term involves using the general term formula directly. The procedure generally requires substituting the known values into the formula to find the specific term you're interested in.
For instance, if you need to calculate the 20th term of the sequence where \(a_1 = -70\) and \(d = -5\), you use the general term formula:
\[ a_{n} = a_{1} + (n - 1) \times d \]Substituting the given values:\[ a_{20} = -70 + (20 - 1) \times (-5) \]Calculating step-by-step:
  • First, simplify inside the parentheses: \((20 - 1) = 19\).
  • Next, multiply by the common difference: \(19 \times (-5) = -95\).
  • Finally, add this result to \(a_1\): \(-70 + (-95) = -165\).
Thus, the 20th term is \(-165\). This step-by-step calculation illustrates how seamlessly the formula can be applied once you have all the necessary values.

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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, \(r\). \(a_{1}=2, r=0.1\)

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(6,3, \frac{3}{2}, \frac{3}{4}, \ldots\)

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=-2, r=-3\)

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 \(2009 .\) $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \text { Year } & \mathbf{2 0 0 0} & \mathbf{2 0 0 1} & \mathbf{2 0 0 2} & \mathbf{2 0 0 3} & \mathbf{2 0 0 4} & \mathbf{2 0 0 5} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 33.87 & 34.21 & 34.55 & 34.90 & 35.25 & 35.60 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|l|l|l|l|} \hline \text { Year } & \mathbf{2 0 0 6} & \mathbf{2 0 0 7} & \mathbf{2 0 0 8} & \mathbf{2 0 0 9} & \mathbf{2 0 1 0} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 36.00 & 36.36 & 36.72 & 37.09 & 37.25 \\ \hline \end{array} $$ 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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