/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 Find the next three terms of eac... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 21

Find the next three terms of each sequence. $$-1.4,2.6,6.6,10.6, \ldots$$

Short Answer

Expert verified
The next three terms are 14.6, 18.6, and 22.6.

Step by step solution

01

Identify the Pattern

First, observe the sequence: \(-1.4, 2.6, 6.6, 10.6\). The terms appear to increase by the same amount each time. Calculate the difference between consecutive terms: \(2.6 - (-1.4) = 4.0\), \(6.6 - 2.6 = 4.0\), and \(10.6 - 6.6 = 4.0\). The common difference is \(4.0\).
02

Determine the Formula for the Sequence

Since it's an arithmetic sequence, the formula for the \(n\)-th term is \(a_n = a_1 + (n-1) imes d\), where \(a_1\) is the first term and \(d\) is the common difference. Here, \(a_1 = -1.4\) and \(d = 4.0\).
03

Calculate Next Terms

Using the formula \(a_n = -1.4 + (n-1) imes 4.0\), calculate the next three terms:- Fifth term: \(a_5 = -1.4 + 4.0 \times 4 = 14.6\).- Sixth term: \(a_6 = -1.4 + 5 imes 4.0 = 18.6\).- Seventh term: \(a_7 = -1.4 + 6 \times 4.0 = 22.6\).

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Key Concepts

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

Common Difference
In an arithmetic sequence, the key feature that defines the sequence is the *common difference*. This is the constant amount that is added to each term to get the next one in the sequence. In our example with the sequence \(-1.4, 2.6, 6.6, 10.6, \ldots\), calculating the common difference involves finding the difference between consecutive terms:
  • \(2.6 - (-1.4) = 4.0\)
  • \(6.6 - 2.6 = 4.0\)
  • \(10.6 - 6.6 = 4.0\)
Since this difference is consistent throughout the sequence, we confirm that the common difference \(d\) is indeed \(4.0\).
This consistent step allows us to easily predict and calculate future terms in the sequence.
Sequence Formula
The formula for finding any term in an arithmetic sequence is crucial once you've identified the common difference. This general formula is given by:
  • \(a_n = a_1 + (n-1) \times d\)
Here, \(a_n\) represents the \(n\)-th term, \(a_1\) is the first term of the sequence, and \(d\) is the common difference.
Applying this to our sequence where \(a_1 = -1.4\) and \(d = 4.0\), the formula becomes:
  • \(a_n = -1.4 + (n-1) \times 4.0\)
This formula provides a clear way to calculate the value of any term we鈥檙e interested in.
Term Calculation
Once the common difference and sequence formula are understood, calculating specific terms becomes straightforward. To find the next terms after \(10.6\) in our given sequence, start with the formula for the \(n\)-th term:
  • *Fifth term*: \(a_5 = -1.4 + 4.0 \times 4 = 14.6\)
  • *Sixth term*: \(a_6 = -1.4 + 5 \times 4.0 = 18.6\)
  • *Seventh term*: \(a_7 = -1.4 + 6 \times 4.0 = 22.6\)
By plugging in consecutive integers for \(n\), we quickly determine the values of the fifth, sixth, and seventh terms. The arithmetic sequence formula makes calculating any terms not just efficient but reliable.

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