Question

write a rule for the nth term of the arithmetic sequence. \(a_8=-15, a_{17}=-78\)

Short answer

The nth term rule of the arithmetic sequence is \(a_n = 34 + (n - 1) * -7\)

Step-by-step solution

Determine the common difference

To find the common difference (d), subtract the 8th term from the 17th term and divide by the difference in their positions. This can be done with this formula \(d = \frac{{a_{17} - a_{8}}}{{17 - 8}}\). Substituting the given values we get: \(d = \frac{{-78 - (-15)}}{{17 - 8}} = -7\)

Calculate first term

Using the formula of an arithmetic series \(a_n = a_1 + (n - 1) * d\), we can rearrange to find the first term \(a_1 = a_n - (n - 1) * d\). We substitute the values of the 8th term and the common difference into the formula and solve for \(a_1\): \(a_1 = -15 - (8 - 1)*(-7) = 34\)

Formulate the nth-term rule

Now that we have \(a_1\) and \(d\), the nth term of the sequence can be given by \(a_n = a_1 + (n - 1) * d = 34 + (n - 1) * -7\)

Key concepts

Common Difference in Arithmetic Sequence

The 'common difference' is a term that comes up often when studying arithmetic sequences. It is the consistent interval between consecutive terms in an arithmetic sequence. More formally, if we consider the consecutive terms of the sequence as \(a_{n}\) and \(a_{n+1}\), the common difference \(d\) would be determined by \(d = a_{n+1} - a_{n}\). This difference remains the same throughout the entire sequence.

For example, in the sequence 2, 5, 8, 11, 14,... the common difference is 3 because each term is 3 more than the previous term. Finding the common difference is crucial because it allows us to predict any term in the sequence without listing out all the terms. In the exercise given, we were able to find \(d\) by using the formula \(d = \frac{{a_{17} - a_{8}}}{{17 - 8}}\), which simplifies the process of defining the nth term rule.

Arithmetic Sequence Formula

The arithmetic sequence formula is a cornerstone in understanding and operating with arithmetic sequences. It is expressed as \(a_n = a_1 + (n - 1) \cdot d\), where \(a_n\) represents the nth term of the sequence, \(a_1\) is the first term, \(n\) is the position of the term in the sequence, and \(d\) is the common difference. This formula provides a direct method to calculate the value of any term in the sequence without having to sequentially list all previous terms.

In the context of the provided exercise, after finding the common difference, we use this formula to identify the first term of the sequence. By rearranging the formula, we can then solve for \(a_1\) as shown in the solution. It is important to note that having either the first term or any specific term, along with the common difference, allows us to determine the position and value of any term within the arithmetic sequence.

Calculating Arithmetic Sequence Terms

Calculating terms in an arithmetic sequence is a straightforward process once the first term and common difference are known. The nth term rule, which was derived from the arithmetic sequence formula, is applied to find any term's value. Using the rule \(a_n = a_1 + (n - 1) \cdot d\), we can plug in the values for \(a_1\), the first term, and \(d\), the common difference, along with \(n\), the term number we wish to find.

For instance, if we need to find the 10th term in a sequence where \(a_1 = 3\) and \(d = 5\), we simply substitute into the nth term rule to get \(a_{10} = 3 + (10 - 1) \cdot 5 = 48\). It's as easy as that! Similarly, in the exercise provided, once we figured out the first term and the common difference, we were able to establish a general rule for finding the nth term, \(a_n = 34 + (n - 1) \cdot -7\), aiding in rapid computation of any term in the sequence.