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Chapter 14: Problem 55

Evaluate \(a+(n-1) d\) for \(a=-2, n=5,\) and \(d=3\)

Short Answer

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Step by step solution

01

Understand the Formula

The formula given is for the nth term of an arithmetic sequence: a + (n-1)dwhere: - 'a' is the first term, - 'n' is the term number, and - 'd' is the common difference.
02

Substitute the Given Values

Substitute the given values into the formula: a = -2, n = 5, and d = 3Thus, the expression becomes: a + (n-1)d = -2 + (5-1)3
03

Simplify Inside the Parentheses

Simplify the expression inside the parentheses (=5-1): -2 + 4脳3
04

Multiply

Multiply the value of the common difference 'd' by the simplified parentheses value: -2 + 4 脳 3 = -2 + 12
05

Final Addition

Further simplifying by adding the final values: -2 + 12 = 10

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Key Concepts

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

nth term formula
In an arithmetic sequence, the nth term formula is a crucial tool that helps you find any term in the sequence. This formula can be written as: \( a + (n-1)d \), where \(a\) represents the first term, \(n\) is the position of the term you want to find, and \(d\) is the common difference. For instance, if you know the first term and the common difference, you can find the 100th term without listing all previous 99 terms.

The nth term formula is especially useful in identifying patterns and making predictions within a sequence. Using this formula, we ensure that we accurately determine any term's value in the sequence.
first term
The first term of an arithmetic sequence, often denoted as \(a\), is the starting point of the sequence. Think of it as the launch pad from which the sequence begins. For example, in the sequence 2, 5, 8, 11, 14..., the first term \(a\) is 2.

In the given exercise, the first term is \(a = -2\). This initial value plays a significant role in determining the subsequent terms when combined with the common difference. By knowing the first term, you set the foundation from which you build the entire sequence.
common difference
The common difference, denoted as \(d\), is a constant value added to each term of the sequence to get the next term. This value can be positive, negative, or zero, dictating whether the sequence is increasing, decreasing, or constant.

In the sequence 2, 5, 8, 11, 14..., the common difference \(d\) is 3. Each term is obtained by adding 3 to the previous term. In the exercise, the common difference \(d = 3\) is used to calculate the fifth term from the first term \(a = -2\) using the nth term formula. This property ensures the sequence progresses in a regular pattern.

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Most popular questions from this chapter

Solve each problem. Constantin Arne has started on a fitness program. He plans to jog 10 min per day for the first week and then add 10 min per day each week until he is jogging an hour each day. In which week will this occur? What is the total number of minutes he will run during the first four weeks?

Evaluate the indicated term for each arithmetic sequence. $$ a_{1}=1, d=-3 ; \quad a_{12} $$

Evaluate the indicated term for each arithmetic sequence. $$ 2,4,6, \ldots ; \quad a_{24} $$

Solve each problem involving an ordinary annuity. A father opened a savings account for his daughter on her first birthday, depositing \(\$ 1000 .\) Each year on her birthday he deposits another \(\$ 1000,\) making the last deposit on her 21 st birthday. If the account pays \(4.4 \%\) interest compounded annually, how much is in the account at the end of the day on the daughter's 21 st birthday? (IMAGE CANNOT COPY)

Solve each problem. Suppose that Cherian Mathew is offered a job at \(\$ 1600\) per month with a guaranteed increase of \(\$ 50\) every six months for 5 yr. What will Cherian's salary be at the end of that time?
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