Arithmetic Sequence Formula Lesson

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. The constant difference used to get from one term to the next, is called the common difference, d. In simpler terms, an arithmetic sequence is one in which you add the same number each time to get the next term in the sequence. You add the common difference, d, to continue finding the next number in the sequence.

The arithmetic sequence formula is used to find any term in the arithmetic sequence without having to keep adding the common difference. This is called finding the nth term of the arithmetic sequence, denoted by an.

The formula is: a_n = a₁ + d(n – 1)

Where

a_n is the nth term of the sequence
a₁ is the first term of the sequence
d is the common difference between consecutive terms
n is the position of the term in the sequence

For example, if you have an arithmetic sequence with the first term 4 and a common difference of 3, to find the 10th term, you would use the formula:

a₁₀ = 4 + 3(10 – 1)

a₁₀ = 4 + 3(9)

a₁₀ = 4 + 27

a₁₀ = 31

To find the common difference of an arithmetic sequence we take the current term (a_n) and subtract the previous term (a_{n – 1}). The formula used to find the common difference is d = a_n – a_{n – 1}.

For example, in the sequence, 3, 5, 7, 9, 11, …, in order to find the common difference we can take the second term (a₂) and subtract the previous term (a₁). Therefore, d = 5 – 7 = 2.

Examples

Example 1: Use the arithmetic sequence formula to find the 12th term in the sequence 2, 5, 8, 11, …

Solution: First verify that the sequence is an arithmetic sequence by determining that the difference is constant between consecutive terms.

Since the difference between each term is 3, we know that this is an arithmetic sequence and that our common difference is 3, d = 3. We also know that our first term is 2, a₁ = 2. Therefore using the arithmetic sequence formula to find the 12th term would be,

a₁₂ = a₁ + d(12 – 1)

a₁₂ = 2 + 3(11)

a₁₂ = 2 + 33

a₁₂ = 35

Therefore, the 12th term of the sequence is 35.

Example 2: Find the first term of the arithmetic sequence if the 22nd term is 172 and the common difference is 8.

Solution: Start by understanding the information that you have and how it fits into the arithmetic sequence formula. Since we know that the 22nd term is 172, we know that a₂₂ = 172, and the common difference is d = 8. Therefore, using the arithmetic sequence formula, we can substitute in the given information and solve for a₁.

a_n = a₁ + d(n – 1)

a₂₂ = a₁ + 8(22 – 1)

172 = a₁ + 8(21)

172 = a₁ + 168

Now we will subtract 168 from both sides of the equation, to obtain the value of the first term.

172 – 168 = a₁ + 168 – 168

4 = a₁

The first term of this arithmetic sequence is 4.

Example 3: Find the common difference, d, if a₁ = 9 and a30 = 328.

Solution: Start by understanding the information that you have and how it fits into the arithmetic sequence formula. We will substitute the given information into the arithmetic sequence formula as shown below.

a₃0 = a₁ + d(30 – 1)

328 = 9 + d(29)

328 – 9 = 9 – 9 + d(29)

319 = d(29)

Now we will divide both sides by 29, to obtain our answer.

319 ÷ 29 = d(29) ÷ 29

11 = d

The common difference for this arithmetic sequence is 11.

FAQs on Arithmetic Sequence Formula

1) What is an arithmetic sequence?

An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the preceding term.

2) How do I find the common difference in an arithmetic sequence?

The common difference (d) in an arithmetic sequence is the constant amount that is added to each term to get the next term. It can be found by subtracting any two consecutive terms in the sequence.

3) Can I find any term in the sequence using the arithmetic sequence formula?

Yes, the arithmetic sequence formula allows you to find any term (an) in the sequence by specifying its position (n).

4) What if I only know two terms of the sequence?

If you know two terms of the sequence (a₁ and a_k) and want to find the common difference (d), you can use the formula:

5) Can an arithmetic sequence have a negative common difference?

Yes, an arithmetic sequence can have a negative common difference. This means that each term is decreasing by a constant amount.

6) How do I know if a sequence is arithmetic?

To determine if a sequence is arithmetic, check if the difference between consecutive terms is constant. If the difference remains the same throughout the sequence, it is arithmetic.

7) Can the first term (a₁) of an arithmetic sequence be negative?

Yes, the first term (a₁) of an arithmetic sequence can be negative. The sequence can start from any real number, positive or negative.