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Chapter 14: Problem 74

Explain how to find the general term of an arithmetic sequence.

Short Answer

Expert verified
The general term of an arithmetic sequence can be found using the formula: \(a_n = a_1 + (n-1) \cdot d\), where \(a_n\) is the nth term, \(a_1\) is the first term, \(n\) is the term number and \(d\) is the common difference.

Step by step solution

01

Identify the First Term

The first term (\(a_1\)) in an arithmetic sequence is the first number in the sequence or series. This value can usually be directly observed or provided within the problem.
02

Calculate the Common Difference

The common difference (\(d\)) is the constant value between each consecutive term. It can be calculated by subtracting the first term from the second term.
03

Formulate the Nth Term

The term to be found (\(a_n\)) is formulated using the arithmetic sequence's formula (\(a_n = a_1 + (n-1) \cdot d\)). Replace \(a_1\), \(d\), and \(n\) with their respective values and compute the result to find \(a_n\), the nth term.

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Beginning at 6: 45 A.M., a bus stops on my block every 23 minutes, so I used the formula for the \(n\) th term of an arithmetic sequence to describe the stopping time for the \(n\) th bus of the day.

Graph each of the functions in the same viewing rectangle. Describe how the graphs illustrate the Binomial Theorem. $$\begin{aligned}&f_{1}(x)=(x+1)^{4} & f_{2}(x)=x^{4}\\\&f_{3}(x)=x^{4}+4 x^{3} & f_{4}(x)=x^{4}+4 x^{3}+6 x^{2}\\\&f_{5}(x)=x^{4}+4 x^{3}+6 x^{2}+4 x\\\&f_{6}(x)=x^{4}+4 x^{3}+6 x^{2}+4 x+1\end{aligned}$$ Use a \([-5,5,1]\) by \([-30,30,10]\) viewing rectangle.

What is a geometric sequence? Give an example with your explanation.

Each exercise involves observing a pattern in the expanded form of the binomial expression \((a+b)^{n}\).$$\begin{array}{l}(a+b)^{1}=a+b \\\\(a+b)^{2}=a^{2}+2 a b+b^{2} \\\\(a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3} \\\\(a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4} \\\\(a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5}\end{array}$$ Describe the pattern for the exponents on \(a\).

$$\text { Simplify: } \sqrt[3]{40 x^{4} y^{7}}$$ (Section \(10.3,\) Example 5 )
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