Question

write a rule for the nth term of the arithmetic sequence. \(a_6=-8, a_{15}=-62\)

Short answer

The rule for the \(n\)th term of the given arithmetic sequence is \(a_n = a_1 + (n - 1) * d\), where \(a_1\) is the first term and \(d\) is the common difference between the terms. The exact values of \(a_1\) and \(d\) can be found by solving the equations from the steps above.

Step-by-step solution

Calculate the Common Difference

Firstly, calculate the common difference, \(d\), using the formula \(d = \frac{(a_n - a_m)}{(n - m)}\). Here, \(a_n\) is the \(n\)th term, \(a_m\) is the \(m\)th term, \(n\) is the position of \(n\)th term and \(m\) is the position of \(m\)th term. Substitute \(n = 15\) , \(m = 6\), \(a_n = -62\) and \(a_m = -8\) into the formula to get \(d\).

Substitute the Common Difference

After calculating \(d\), substitute it into the formula \(a_n = a_1 + (n - 1) * d\) to find \(a_1\). \(a_n\) is the \(n\)th term, \(n\) is the position of \(n\)th term and \(d\) is the common difference. For \(n = 6\) and \(a_n = -8\), substitute these values into the equation.

Find the First Term

Solve the equation from Step 2 to determine \(a_1\), the first term of the sequence.

Formulate the Rule

Finally, use the values of \(a_1\) and \(d\) found in the previous steps to set the rule of the \(n\)th term of the sequence. The rule for the \(n\)th term in an arithmetic sequence is \(a_n = a_1 + (n - 1) * d\).

Key concepts

Common Difference

Understanding the common difference is crucial in studying arithmetic sequences. This value, denoted as \(d\), is the consistent interval between consecutive terms of an arithmetic sequence. In simpler terms, it's how much you add to (or subtract from) one term to get the next.
To calculate the common difference, take any two terms in the sequence where their positions are known, and use the formula \(d = \frac{(a_n - a_m)}{(n - m)}\). If you subtract the lesser term's value from the greater term's value and divide by the distance between their positions in the sequence, you'll uncover this constant rate of change.
For instance, given the terms \(a_6=-8\) and \(a_{15}=-62\), we can calculate the common difference by subtracting the value of the sixth term from the fifteenth term, and then dividing by the number of terms between these positions, which is \(15 - 6\). This step establishes a foundation for determining all other elements of the sequence.

Nth Term

The nth term of an arithmetic sequence is a formula that allows you to find any term in the sequence, given its position, \(n\), within that sequence. This agreement eliminates the need to list all terms up to the one you're interested in.
To find the nth term, you should know the first term, \(a_1\), and the common difference, \(d\). The formula for the nth term is \(a_n = a_1 + (n - 1) * d\). It expresses each term as a function of its position in the sequence.
Once the common difference is known, you can use it alongside any term of the sequence to calculate the first term. With the first term and the common difference at hand, you can construct the rule for the nth term, which represents a general instruction for finding any term's value based on its position. In the given exercise, this process includes substituting the values into the formula and solving for \(a_1\) using one of the known terms.

Arithmetic Series

An arithmetic series is the sum of the terms in an arithmetic sequence. If you're picturing a long addition problem that combines each term in the sequence, you have the right idea. It's useful when dealing with problems requiring the sum of a series of numbers with a common difference.
To calculate an arithmetic series, you can use different formulas, but one of the simplest involves knowing the first term, the last term, and the number of terms in the series. The formula is:\[ S_n = \frac{n}{2} * (a_1 + a_n) \], where \(S_n\) is the sum of the first \(n\) terms, \(a_1\) is the first term, \(a_n\) is the nth, or last, term, and \(n\) is the number of terms.
When solving for the sum of an arithmetic series it is important to have already identified the sequence's common difference and nth term, as these allow you to quickly find any term and accurately calculate the total sum. This concept is essential for understanding patterns in number sequences and applying them to various mathematical scenarios, from simple addition to complex financial calculations.