Question

Find the indicated term in each arithmetic sequence. $$ 80 \text { th term of } 5,0,-5, \ldots $$

Short answer

-390

Step-by-step solution

Identify the first term and common difference

The first term of the arithmetic sequence is denoted by \(a_1\). Here, \(a_1 = 5\). The common difference, denoted by \(d\), is the difference between consecutive terms. \(d = 0 - 5 = -5\).

Use the formula for the nth term of an arithmetic sequence

The formula for the nth term of an arithmetic sequence is given by: \(a_n = a_1 + (n-1)d\).

Substitute values into the formula

Substitute the known values into the formula where \(n = 80\), \(a_1 = 5\), and \(d = -5\): \(a_{80} = 5 + (80-1)(-5)\).

Calculate the expression inside the parentheses

First compute \(80-1 = 79\).

Multiply by the common difference

Now, multiply by the common difference: \(79 \times (-5) = -395\).

Add the result to the first term

Finally, add the result to the first term: \(5 + (-395) = -390\).

Key concepts

arithmetic sequence

An arithmetic sequence, also known as an arithmetic progression, is a list of numbers where each term after the first is obtained by adding a constant difference to the previous one. This constant difference is referred to as the 'common difference'.
In an arithmetic sequence, the terms grow (or shrink) in a linear manner because of this fixed difference. For example, in the sequence 5, 0, -5, -10, the common difference is -5.
To identify an arithmetic sequence, note the pattern among the numbers. If the difference between every consecutive term is consistent, you have an arithmetic sequence.

common difference

The common difference in an arithmetic sequence is the fixed number that you add (or subtract) to get from one term to the next. It’s denoted by the letter 'd'.
To find the common difference, subtract any term from the one that follows it. For the sequence given in the exercise, 5, 0, -5,..., we calculate the common difference as follows:
\(d = 0 - 5 = -5\)
So, the common difference here is -5.
The common difference in an arithmetic sequence is crucial as it directly influences the rate at which the terms of the sequence change.

nth term formula

The nth term formula of an arithmetic sequence gives us a way to find any term in the sequence without having to list all the terms. The formula is given by:
\[a_n = a_1 + (n-1) \times d\]
where:

• \(a_n\) is the nth term of the sequence
• \(a_1\) is the first term of the sequence
• \(n\) is the position of the term we want to find
• \(d\) is the common difference

For this exercise, we use the values: \(a_1 = 5\), \(n = 80\), and \(d = -5\).
This formula simplifies finding terms in the sequence without manually adding the common difference repeatedly.

sequence solving

Solving problems involving arithmetic sequences often involves a few clear steps:

• Identify the first term, \(a_1\), and the common difference, \(d\).
• Write down the nth term formula: \(a_n = a_1 + (n-1) \times d\).
• Substitute the given values into the formula.
• Perform the arithmetic operations to solve for the nth term.

Let’s apply these steps to our problem. Here, we aim to find the 80th term of the sequence 5, 0, -5, ...
- First term, \(a_1 = 5\)
- Common difference, \(d = -5\)
- We want the 80th term, \(a_{80}\)
Using the formula:
- Substitute the values: \(a_{80} = 5 + (80-1) \times (-5)\)
- Simplify inside the parentheses: \(80-1 = 79\)
- Multiply: \(79 \times (-5) = -395\)
- Add to the first term: \(5 + (-395) = -390\)
The result is that the 80th term is -390.