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Chapter 10: Problem 5

Find the terms \(a_{0}, a_{1},\) and \(a_{2}\) for each sequence. $$a_{n}=4+6 n$$

Short Answer

Expert verified
The terms \(a_{0}, a_{1},\) and \(a_{2}\) of the sequence are 4, 10, and 16 respectively.

Step by step solution

01

Calculate the term when \(n=0\)

Substitute \(n = 0\) into the sequence formula \(a_{n}=4+6 n\), this gives \(a_{0} = 4 + 6 \cdot 0 = 4\).
02

Calculate the term when \(n=1\)

Substitute \(n = 1\) into the sequence formula \(a_{n}=4+6 n\), this gives \(a_{1} = 4 + 6 \cdot 1 = 10\).
03

Calculate the term when \(n=2\)

Substitute \(n = 2\) into the sequence formula \(a_{n}=4+6 n\), this gives \(a_{2} = 4 + 6 \cdot 2 = 16\).

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Key Concepts

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

Arithmetic Sequence
An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. Imagine a staircase where each step is the same height. Similarly, in an arithmetic sequence, each term is obtained by adding the same fixed number to the previous term. This fixed number is called the common difference, often denoted by \(d\). The sequence can be represented by the general formula:
\[a_n = a_1 + (n-1) \, d\]
where \(a_n\) is the \(n^{th}\) term, \(a_1\) is the first term, and \(n\) is the position of the term in the sequence. The formula helps us determine any term in the sequence without listing all preceding terms.
Term Calculation
Calculating terms in a sequence involves substituting the term's position number into the sequence formula. Let's explore how it works using the specific formula given in the task: \(a_n = 4 + 6n\).
For the term calculation, ensure:
  • You know the formula for the sequence.
  • Identify which term (\(n\)) you're finding, like \(n = 0\), \(n = 1\), or \(n = 2\).
  • Substitute \(n\) into the formula to find \(a_n\).
This substitution will allow you to calculate specific terms directly. For instance, setting \(n = 0\) gives the zero term \(a_0 = 4 + 6 \times 0 = 4\), while \(n = 1\) results in \(a_1 = 4 + 6 \times 1 = 10\), and \(n = 2\) results in \(a_2 = 4 + 6 \times 2 = 16\). Calculating each value separately provides the terms: 4, 10, and 16.
Formula Substitution
Formula substitution is a key skill when working with sequences. It means using the given formula and substituting specific values to find the result. This technique applies beyond just finding sequence terms; it's widely used across many math concepts.
To effectively use formula substitution:
  • Read the formula carefully and identify what each symbol represents. In our case, \(a_n = 4 + 6n\), \(a_n\) is the term we aim to find, and \(n\) is the sequence position.
  • Simply replace \(n\) with the given or desired value. For example, substitute \(n=1\) to find \(a_1\).
  • After substituting the value, simplify the equation to find the term.
Through substitution, the abstract formula becomes a concrete tool, helping you work through problems methodically and ensuring accuracy in calculations.

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Most popular questions from this chapter

Induction is not the only method of proving that a statement is true. Exercises \(26-29\) suggest alternate methods for proving statements. By factoring \(a^{3}-b^{3}, a\) and \(b\) positive integers, show that \(a^{3}-b^{3}\) is divisible by \(a-b\)

Concepts This set of exercises will draw on the ideas presented in this section and your general math background. \- A sequence \(b_{0}, b_{1}, b_{2}, \ldots\) has the property that \(b_{n}=\) \(\left(\frac{n+3}{n+2}\right) b_{n-1}\) for \(n=1,2,3, \ldots,\) where \(c\) is a positive constant to be determined. Find \(c\) if \(b_{2}=25\) and \(b_{4}=315\)

In Exercises \(5-25,\) prove the statement by induction. \(n^{2}+3 n\) is divisible by 2

Assume that the probability of winning 5 dollars in the lottery (on one lottery ticket) for any given week is \(\frac{1}{50},\) and consider the following argument. "Henry buys a lottery ticket every week, but he hasn't won 5 dollars in any of the previous 49 weeks, so he is assured of winning 5 dollars this week." Is this a valid argument? Explain.

Answer True or False. Consider randomly picking a card from a standard deck of 52 cards. The complement of the event "picking a black card" is "picking a heart."
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