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Chapter 14: Problem 19

Find a general term \(a_{n}\) for the given terms of each sequence. $$ 4,8,12,16, \dots $$

Short Answer

Expert verified
The general term is \(a_{n} = 4n\).

Step by step solution

01

Identify the first term

Identify the first term of the sequence. The first term is given as 4.
02

Determine the common difference

Find the difference between consecutive terms. For this sequence: \(8 - 4 = 4\), \(12 - 8 = 4\), and \(16 - 12 = 4\). Therefore, the common difference is 4.
03

Write the general term formula for an arithmetic sequence

The general term for an arithmetic sequence is given by the formula: \(a_{n} = a_1 + (n-1) \times d\), where \(a_1\) is the first term and \(d\) is the common difference.
04

Plug in the values

Substitute the first term \(a_1 = 4\) and common difference \(d = 4\) into the general term formula: \(a_{n} = 4 + (n-1) \times 4\).
05

Simplify the expression

Distribute and combine like terms: \[a_{n} = 4 + 4(n-1)\] \[a_{n} = 4 + 4n - 4\] \[a_{n} = 4n\].

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Key Concepts

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

Arithmetic Sequence
In mathematics, an arithmetic sequence is a list of numbers where each term after the first is found by adding a constant, known as the common difference, to the previous term.
For example, in the sequence 4, 8, 12, 16, ..., the difference between consecutive terms is always 4.
Arithmetic sequences are crucial because they are simple and predictable patterns, making it easier to find any term in the sequence without listing all the previous terms.
If you know the first term and the common difference, you can easily build the whole sequence.
Common Difference
The common difference in an arithmetic sequence is the value that you add (or subtract) to get from one term to the next.
Finding the common difference is straightforward: subtract any term from the term that follows it.
For our example sequence 4, 8, 12, 16:
  • 8 - 4 = 4
  • 12 - 8 = 4
  • 16 - 12 = 4
This consistent value (4 in our case) is the common difference.
Understanding the common difference is key for describing how the sequence grows or changes.
General Term Formula
The general term formula for an arithmetic sequence allows us to find any term without listing all the terms before it.
The formula is expressed as: !MathLatex![a_n = a_1 + (n-1) \times d].
Here:
  • \(a_n\) is the nth term you're trying 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.
For our sequence:
Since the first term \(a_1\) is 4 and the common difference \(d\) is 4, we can substitute these values into the formula: !MathLatex![a_n = 4 + (n-1) \times 4]
Simplifying the expression: !MathLatex![a_n = 4 + 4(n-1]
Which is !MathLatex![a_n = 4n].
So, the general term of our sequence is 4 times the position of the term in the sequence!

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Most popular questions from this chapter

Use a formula for \(S_{n}\) to evaluate each series. $$ \sum_{i=1}^{17}(3 i-1) $$

Find a general term \(a_{n}\) for the given terms of each sequence. $$ 7,14,21,28, \dots $$

Solve each applied problem by writing the first few terms of a sequence. A certain car loses \(\frac{1}{2}\) of its value each year. If this car cost \(\$ 40,000\) new, what is its value at the end of 6 yr?

If the given sequence is geometric, find the common ratio \(r\). If the sequence is not geometric, say so. See Example 1 . $$ 4,8,16,32, \dots $$

If the given sequence is geometric, find the common ratio \(r\). If the sequence is not geometric, say so. See Example 1 . $$ \frac{5}{7}, \frac{8}{7}, \frac{11}{7}, 2, \dots $$
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