Download Lesson 34, Recognizing and Extending Arithmetic Sequences

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Transcript 
Warm Up
1. Any quantity whose value does not change is called a __________________.
2. Simplify 7.2 − 5.8 − (−15)
3. Simplify −0.12 − −43.7 − 73.5
4. Simplify 6(−2.5)
5. Simplify (−15)(−4.2)
Lesson 34: Recognizing and
Extending Arithmetic Sequences
The number system, expressions and equations
Vocabulary
• Sequence: a list of numbers that follow a rule
• Term of a sequence: a number in the sequence
• Arithmetic sequence: a sequence that has a constant difference between two
consecutive terms
• Common difference: the constant difference between two consecutive terms of a
sequence
Arithmetic Sequence Formula
Arithmetic Sequence Formula
Use the formula below to find the next term in a sequence:
𝑎𝑛 = 𝑎𝑛−1 + 𝑑
𝑎1 = 𝑓𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚
𝑑 = 𝑐𝑜𝑚𝑚𝑜𝑛 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒
𝑛 = 𝑡𝑒𝑟𝑚 𝑛𝑢𝑚𝑏𝑒𝑟
In the arithmetic sequence 7, 11, 15, 19, … , 𝑎1 = 7, 𝑎2 = 11, 𝑎3 = 15, 𝑎𝑛𝑑 𝑎4 =
19. The common difference is 4.
Finding a formula for the nth term
• We can find a formula for the nth term by looking more closely at the
sequence 7, 11, 15, 19, …
Term
Term Number (𝑛)
Sequence Pattern
Description
1st or 𝑎1
1
7
𝑎1
2nd or 𝑎2
2
7 + 4 = 7 + (1)4
𝑎1 + 1 𝑑
3rd or 𝑎3
3
7 + 4 + 4 = 7 + (2)4
𝑎1 + 2 𝑑
4th or 𝑎4
4
7+4+4+4=7+ 3 4
𝑎1 + 3 𝑑
5th or 𝑎5
5
7+4+4+4+4= 7+ 4 4
𝑎1 + 4 𝑑
𝑛𝑡ℎ or 𝑎𝑛
𝑛
7 + (𝑛 − 1)4
𝑎1 + 𝑛 − 1 𝑑
Finding the nth term
Finding the 𝑛th Term of an Arithmetic Sequence
𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑
𝑎1 = 𝑓𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚
𝑑 = 𝑐𝑜𝑚𝑚𝑜𝑛 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒
Example
• Determine if each sequence is an arithmetic sequence. If yes, find
the common difference.
a. 7, 12, 17, 22, …
b. 3, 6, 12, 24, …
Example
• Use a recursive formula to find the first four terms of an arithmetic
sequence where 𝑎1 = −2 and the common difference 𝑑 = 7.
Example
a. Use the rule 𝑎𝑛 = 6 + 𝑛 − 1 2 to find the 4th and 11th terms of the
sequence.
b. Find the 10th term of the sequence 3, 11, 19, 27, …
c. Find the
10th
term of the sequence
1 3 5 7
, , , ,…
4 4 4 4
Example
• The first table at a reception will seat 9 guests while each additional
table will seat 6 more guests.
a. Write a rule to model this situation.
b. Use the rule to find how may guests can be seated with 10 tables.
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