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Chapter 8: Problem 16

, write a rule for the nth term of the sequence. Then fi nd a20. \(86,79,72,65, \ldots\)

Short Answer

Expert verified
The rule for the nth term of the sequence is \(a_n = 86 + (n-1) * -7\). The 20th term of the sequence is -47.

Step by step solution

01

Identify the pattern

For an arithmetic sequence, the difference between consecutive terms is consistent. In this case, every subsequent number subtracts 7 from the previous one. So, the common difference (d) is -7.
02

Write a formula for the nth term

The formula for the nth term (a_n) of an arithmetic sequence is given as \(a_n = a_1 + (n-1) * d\), where \(a_1\) is the first term and d is the common difference. In this sequence, \(a_1 = 86\) and \(d = -7\). Therefore, the nth term can be expressed as \(a_n = 86 + (n-1) * -7\).
03

Substitute n with 20

If asked for the 20th term (a_20), plug in 20 for n in the formula: \(a_{20} = 86 + (20-1) * -7\).
04

Calculate the Value

The calculation results in \(a_{20} = 86 - 19 * 7\), which simplifies to \(a_{20} = 86 - 133 = -47\).

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Key Concepts

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

Understanding Common Difference
In an arithmetic sequence, the common difference is the consistent interval between consecutive terms. It's derived by subtracting the first term from the second, or any term from the one immediately before it. For instance, in the sequence provided, 86, 79, 72, 65, and so on, you subtract 79 from 86, resulting in -7. This -7 is the common difference. It's important because it dictates the pattern of the sequence and helps in constructing further terms.
For every arithmetic sequence, identifying this difference is a key first step. Always check whether the same difference applies as you move through several terms, ensuring the sequence is indeed arithmetic.
Deriving the nth Term Formula
The nth term formula is crucial in finding any term in an arithmetic sequence without listing all the terms. Given by the expression:
\(a_n = a_1 + (n-1) \cdot d\)
where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number being calculated. In our example, \(a_1 = 86\) and \(d = -7\). By substituting these values, we get:
\(a_n = 86 + (n-1) \cdot (-7)\).
This formula acts as a universal rule to determine any term in the sequence, empowering us to solve for challenges like finding the 20th term or any other.
Exploring Arithmetic Progression
An arithmetic progression (AP) is a type of sequence where each term after the first is found by adding the common difference to the previous term. It creates a linear pattern with a steady rate of increase or decrease.
For example, in a sequence such as 86, 79, 72, 65, and so on, the pattern reveals a decrement by 7 each time. This arithmetic progression keeps the flow methodical and predictable, often visualized as points on a straight line if plotted on a graph. Recognizing this pattern leads to effective predictions of future terms.
Detecting Sequence Patterns
Sequence pattern detection is a significant skill when dealing with arithmetic sequences. It involves identifying the consistent intervals or differences between the sequence elements, which in turn reveals the sequence type. In the sequence 86, 79, 72, 65, we can observe that each term decreases consistently by 7.
By spotting this, you learn that the sequence follows an arithmetic pattern, specifically an arithmetic progression. This skill of detecting patterns not only applies to numerical sequences but to complex problem-solving scenarios, like coding or data analysis, where spotting regularities can lead to simplified solutions.

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Most popular questions from this chapter

In Exercises 33-40, write a rule for the \(n\)th term of the geometric sequence. $$ a_1=1, a_2=49 $$

REASONING Find the sum of the positive odd integers less than 300 . Explain your reasoning.

Write an explicit rule for the sequence. $$ a_1=13, a_n=4 a_{n-1} $$

What is the 873rd term of the sequence whose first term is \(a_1=0.01\) and whose \(n\)th term is \(a_n=1.01 a_{n-1}\) ? Justify your answer. (A) \(58.65\) (B) \(8.73\) (C) \(1.08\) (D) \(586,459.38\)

In the puzzle called the Tower of Hanoi, the object is to use a series of moves to take the rings from one peg and stack them in order on another peg. A move consists of moving exactly one ring, and no ring may be placed on top of a smaller ring. The minimum number \(a_n\) of moves required to move \(n\) rings is 1 for 1 ring, 3 for 2 rings, 7 for 3 rings, 15 for 4 rings, and 31 for 5 rings. a. Write a rule for the sequence. b. What is the minimum number of moves required to move 6 rings? 7 rings? 8 rings?
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