Arithmetic Series

Arithmetic Sequences

By Steven Giesting

Multiple Select

Which of the following sequences are arithmetic?

• 5, 10, 15, 20, . . .

• 5, 10, 20, 40, . . .

• 3, 5, 7, 9, . . .

• 1, 1, 2, 3, 5, 8, . . .

​A recursive definition of an arithmetic sequence has 2 parts:

a₁ = a (the first term in the sequence)

a_n = a_{n -1}  + d, for n > 1 (the recursive formula)

A recursive definition lets us find any term of the sequence if we know the previous terms.

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If we look at our previous example, 5, 10, 15, 20, . . .

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A recursive definition would be: a_n=a_n-1 + 5​

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Recursive definitions would be good for writing spreadsheet formulas​

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​	​A	B
​1	5​	​
2​	10​	​
3​	15​	​
4​	20​	​
5​	​=a4+5	​
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​	​	​
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​An explicit definition for an arithmetic sequence is a single formula:

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a_n = a₁ + (n - 1)d, for n > 1  

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If you think about it, the graph of an arithmetic sequence is the slope-intercept form.​

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​Example 1: Write an explicit definition for the following sequence 3, 5, 7, 9

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a_n = 3 + (n - 1)2

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a_n = 3 + 2n - 2

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a_n = 2n + 1​

​The arithmetic mean, or average, of two numbers is (x +y)/2

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3, 18, 33, 48, . . .

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Find the arithmetic mean of the 1st and 3rd numbers?

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(3 + 33)/2 = 36/2 = 18

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What do you notice?

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Try it for the 2nd & 4th numbers

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(18 +48)/2 = 66/2 = 33​

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​What is the missing term of the arithmetic sequence 

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. . . 35, ____, 53, . . .

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(35 + 53)/2 = 88/2 = 44, d = 9​

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​The 9th and 11th terms of an arithmetic sequence are 132 and 98. 

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What is the 10th term?

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(132 + 98)/2 = 230/2 = 115, d = -17​

​The number of seats in the first 16 rows in a curved section of an arena form an arithmetic sequence.  If there are 20 seats in Row 1 and 23 seats in Row 2, how many seats are in Row 16?

​1) Find the explicit definition:

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d = 3, so a_n = 20 + (n-1)3 = 20 + 3n - 3 = 3n + 17

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2) Use the explicit definition:

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​ a_n= 3n + 17 = 3(16) + 17 = 48 + 17 = 65