4.7 Arithmetic Sequences

• A sequence is a set of numbers in a specific order.

• The numbers in the sequence are called terms.

• If the difference between successive terms is constant, then it is called an arithmetic sequence.

• The difference between the terms is called the common difference.

Arithmetic Sequence

• An arithmetic sequence is a numerical pattern that increases or decreases at a constant rate or value called the common difference.

Identify Arithmetic Sequences

• Determine whether each sequence is arithmetic. Justify your answer.

a. 1, 2, 4, 8, …

This is not an arithmetic sequence because the difference between terms is not constant.

Identify Arithmetic Sequences

• Determine whether each sequence is arithmetic. Justify your answer.

b. ½, ¼, 0, -¼ , …

This is an arithmetic sequence because the difference between terms is constant. The difference is decrease by ¼ .

Writing Arithmetic Sequences

• You can use the common difference of an arithmetic sequence to find the next term in the sequence.

• Each term of an arithmetic sequence after the first term can be found by adding the common difference to the preceding term.

Extend a Sequence

• Find the next three terms of the arithmetic sequence 74, 67, 60, 53, …

• Find the common difference by subtracting successive terms.

• The common difference is -7.

• Add -7 to the last term of the sequence to get the next term in the sequence.

• The next three terms are 46, 39, and 32.

Arithmetic Sequences

Nth term of an Arithmetic Sequence

Find a Specific Term

• Find the 14th term in the arithmetic sequence 9, 17, 25, 33, …

• Find the common difference.

• The common difference is 8.

• Use the formula for the nth term of an arithmetic sequence.

Find a Specific Term

• Find the 14th term in the arithmetic sequence 9, 17, 25, 33, …

an = a1 + (n – 1) d

a14 = 9 + (14 – 1) 8

a14 = 9 + 104

a14 = 113

Write an Equation for a Sequence

• Consider the arithmetic sequence 12, 23, 34, 45, …

a. Write an equation for the nth term of the sequence.

In this sequence, the first term is 12. find the common difference.

The common difference is 11.Use the formula for the nth term to

write an equation.

Write an Equation for a Sequencean = a1 + (n – 1) d

an = 12 + (n – 1) 11

an = 12 + 11n – 11

an = 11n + 1

b. Find the 10th term in the sequence.

an = 11n + 1

a10 = 11(10) + 1

a10 = 111