3.10: Arithmetic Sequences

Identifying Arithmetic Sequences

A sequence of numbers is just that, a list of numbers in order. It can be a short list, such as the number of points earned on each assignment in a class, such as {10, 10, 8, 9, 10, 6, 10}. Or it can be a longer list, even infinitely long, such as the list of prime numbers. For example, here’s a sequence of numbers, specifically, the squares of the first 12 natural numbers.

{1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144}

Each value in the sequence is called a term. Terms in the list are often referred to by their location in the sequence, as in the

The notation we use with sequences is a letter, which represents a term in the sequence, and a subscript, which indicates what place the term is in the sequence. For the sequence {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144}, we will use the letter as a value in the sequence, and so would be the term in the sequence at the fifth position. That number is 25, so we can write .

In this section, we focus on a special kind of sequence, one referred to as an arithmetic sequence. Arithmetic sequences have terms that increase by a fixed number or decrease by a fixed number, called the constant difference (denoted by ), provided that value is not 0. This means the next term is always the previous term plus or minus a specified, constant value. Another way to say this is that the difference between any consecutive terms of the sequence is always the same value.

To see a constant difference, look at the following sequence: {7, 15, 23, 31, 39, 47, 55, 63, 71, 79, 87}. Figure 3.49 illustrates that each term of the sequence is the previous term plus 8. Eight is the constant difference here.

Arithmetic sequences can be expressed with a formula. When we know the first term of an arithmetic sequence, which we label , and we know the constant difference, which is denoted , we can find any other term of the arithmetic sequence. The formula for the term of an arithmetic sequence is .

Let’s examine the formula with this arithmetic sequence: . In this sequence and . The table below shows the values calculated.

We can see how the term can be directly calculated. In this sequence, the formula is where the first term, , is 4 and the constant difference is 3. We can then determine the term of this sequence: .

If we know two terms of the sequence, it is possible to determine the general form of an arithmetic sequence, .

Finding the Sum of a Finite Arithmetic Sequence

Sometimes we want to determine the sum of the numbers of a finite arithmetic sequence. The formula for this is fairly straightforward.

Using Arithmetic Sequences to Solve Real-World Applications

Applications of arithmetic sequences occur any time some quantity increases by a fixed amount at each step. For instance, suppose someone practices chess each week and increases the amount of time they study each week. The first week the person practices for 3 hours, and vows to practice 30 more minutes each week. Since the amount of time practicing increases by a fixed number each week, this would qualify as an arithmetic sequence.

Check Your Understanding

1. Is the following an arithmetic sequence? Explain.

{3, 6, 9, 15, 25, 39, 90}

2. What is the 7th term of the following sequence?

{1, 5, 7, 100, 4, -17, 8, 100, 19, 7.6, 345}

3. In an arithmetic sequence, the first term is 10 and the constant difference is 4.5. What is the 135th term?

4. If the eighth term of an arithmetic sequence is 35 and the 40th term is 131, what is the constant difference and the first term of the sequence?

5. What is the sum of the first 100 terms of the arithmetic sequence with first term 4 and constant difference 7?

6. A new marketing firm began with 30 people in its survey group. The firm adds 4 people per day. How many people will be in their survey group after 100 days?