Arithmetic Sequences: Definition, Formula, and Examples

What is an Arithmetic Sequence?

An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is known as the common difference. For example, the sequence 2, 5, 8, 11, 14 is an arithmetic sequence with a common difference of 3.

Formula for Arithmetic Sequences

The general formula for an arithmetic sequence is:

a_n = a₁ + (n - 1)d

where:

• a_n is the nth term of the sequence
• a₁ is the first term of the sequence
• d is the common difference
• n is the term number

Example

Let's say we have an arithmetic sequence with a first term of 5 and a common difference of 2. What is the 10th term of this sequence?

Using the formula, we can find the 10th term:

a₁₀ = 5 + (10 - 1)2

a₁₀ = 5 + 18

a₁₀ = 23

Therefore, the 10th term of the sequence is 23.

Applications of Arithmetic Sequences

Arithmetic sequences have many applications in real life. For example, they can be used to model:

• The growth of a population
• The depreciation of an asset
• The amount of money in a savings account that earns simple interest

Practice Problems

Here are a few practice problems to test your understanding of arithmetic sequences:

1. Find the common difference of the arithmetic sequence 7, 11, 15, 19, ...
2. Find the 15th term of the arithmetic sequence 2, 6, 10, 14, ...
3. A car depreciates in value by $1,000 each year. If the car is initially worth $20,000, what will its value be after 5 years?

Answers

1. The common difference is 4.
2. The 15th term is 58.
3. The car will be worth $15,000 after 5 years.