Draw the next three shapes in the pattern

1.

2.

3.Can You Find the Pattern?

4. 20, 16, 12, 8, ___, ___, __

5. -9, -4, 1, 6, ___, ___, ___

6. 1, 10, 100, ___, ___, ___

7. 3, 5, 1, 3, -1, ___, ___, ___

4, 0, -4

11, 16, 21

1000, 10000, 100000

1, -3, -1

Unit 6 - SequencesUnit 6 - SequencesVocabulary

Arithmetic Sequence

ObjectivesRecognize and extend an arithmetic sequence.

Find a given term of an arithmetic sequence.

PatternSequenceTermArithmetic SequenceCommon DifferenceConsecutive

Vocabulary

DefinitionsDefinitions

Sequence – a list of numbers that often form a pattern

Term – an element or number in a sequence

Arithmetic Sequence – a sequence whose successive terms differ by the same nonzero number, d, called the common difference

Common Difference – In arithmetic sequence, the nonzero constant difference of any term and the previous term

(x1, x2, x3,…) (a, ar, ar2, ar3,…)

Sequence(0, 1, 1, 2, 3, 5,…) (C, R, Y,

G)

Pattern 0, 5, 10, 15,20,…

-6, -2, 2, 6, 10,… 36, 30, 24, 18, 12,

…

Arithmetic sequenceYes: 2, 5, 8, 11, 14, 17, 20…

No: 0, 1, 1, 2, 3, 5, 8, 13…Yes: 4, 8, 12, 16, 20, 24, 28…

During a thunderstorm, you can estimate your distance from a lightning strike by counting the number of seconds from the time you see the lightning until you hear the thunder.

When you list the times and distances in order, each list forms a sequence. A sequence is a list of numbers that often forms a pattern. Each number in a sequence is a term.

Distance (mi)

1 542 6 7 83

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Time (s)

+0.2 +0.2 +0.2 +0.2 +0.2 +0.2 +0.2

Notice that in the distance sequence, you can find the next term by adding 0.2 to the previous term. When the terms of a sequence differ by the same nonzero number d, the sequence is an arithmetic sequence and d is the common difference. So the distances in the table form an arithmetic sequence with the common difference of 0.2.

Time (s)

Distance (mi)

Identifying Arithmetic Sequences

Determine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms.

9, 13, 17, 21,…Step 1 Find the difference between successive terms.

You add 4 to each term to find the next term. The common difference is 4.

9, 13, 17, 21,…+4 +4 +4

Step 2 Use the common difference to find the next 3 terms.

9, 13, 17, 21,

+4 +4 +4

The sequence appears to be an arithmetic sequence with a common difference of 4. The next 3 terms are 25, 29, 33.

25, 29, 33,…

Reading Math

The three dots at the end of a sequence are called an ellipsis. They mean that the sequence continues and can read as “and so on.”

Identifying Arithmetic Sequences

Determine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms.

10, 8, 5, 1,…

Find the difference between successive terms.

The difference between successive terms is not the same.

This sequence is not an arithmetic sequence.

10, 8, 5, 1,…

–2 –3 –4

Determine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms.

Try This!

–4, –2, 1, 5,…

Step 1 Find the difference between successive terms.

–4, –2, 1, 5,…

+2 +3 +4

The difference between successive terms is not the same.

This sequence is not an arithmetic sequence.

4, 1, –2, –5,…

Step 1 Find the difference between successive terms.

You add –3 to each term to find the next term. The common difference is –3.

4, 1, –2, –5,…

–3 –3 –3

Determine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms.

Try This!

Step 2 Use the common difference to find the next 3 terms.

4, 1, –2, –5,

The sequence appears to be an arithmetic sequence with a common difference of –3. The next three terms are –8, –11, –14.

–8, –11, –14,… –3 –3 –3

The variable a is often used to represent terms in a sequence. The variable a9, read “a sub 9,” is the ninth term, in a sequence. To designate any term, or the nth term in a sequence, you write an, where n can be any number.

1 2 3 4… n Position

The sequence above starts with 3. The common difference d is 2. You can use the first term and the common difference to write a rule for finding an.

3, 5, 7, 9… Terma1 a2 a3 a4 an

The pattern in the table shows that to find the nth term, add the first term to the product of (n – 1) and the common difference.

n = nth term or term you are looking for

= 1st term in the sequenced = common difference

a1

Try This!

Find the indicated term of the arithmetic sequence.

60th term: 11, 5, –1, –7, …

Step 1 Find the common difference.

11, 5, –1, –7,…

–6 –6 –6

The common difference is –6.

Step 2 Write a rule to find the 60th term.

The 60th term is –343.

Write a rule to find the nth term.

Simplify the expression in parentheses.

Multiply.

Add.

Substitute 11 for a1, 60 for n, and –6 for d.

an = a1 + (n – 1)d

a60 = 11 + (60 – 1)(–6)

= 11 + (59)(–6)

= 11 + (–354)

= –343

Try This!

Find the indicated term of the arithmetic sequence.

12th term: a1 = 4.2; d = 1.4

Write a rule to find the nth term.

Simplify the expression in parentheses.

Multiply.

Add.

The 12th term is 19.6.

Substitute 4.2 for a1,12 for n, and 1.4 for d.

an = a1 + (n – 1)d

a12 = 4.2 + (12 – 1)(1.4)

= 4.2 + (11)(1.4)

= 4.2 + (15.4)

= 19.6