Question

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

Short answer

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

Step-by-step solution

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 \(4.0\).

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 \(d = 4.0\).

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: \(a_7 = -1.4 + 6 \times 4.0 = 22.6\).

Key concepts

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’re 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.