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10-03 Arithmetic Sequences and Series
Summary: In this section, you will:
Write the explicit rule for an arithmetic sequence.
Write the recursive rule for an arithmetic sequence.
Evaluate the sum for an arithmetic series.
SDA NAD Content Standards (2018): PC.7.2
You are saving money to buy your first car. You save $100 every month with a goal of $3000. That means your total money is $100 the first month, $200 the second month, $300 the third month, etc. These totals are an arithmetic sequence.
Arithmetic Sequences
An arithmetic sequence with a pattern of a common difference, d. We can derive the explicit formula by looking at a simple example.
$$ 1, 3, 5, 7, 9, … $$
Write the numbers using the pattern which in this example is adding 2.
It appears that the common difference is 3, so \(d = 3\).
$$ a_n = a_1 + (n - 1)d $$
$$ a_n = 4 + (n - 1)3 $$
Simplify.
$$ a_n = 4 + 3n - 3 $$
$$ a_n = 3n + 1 $$
Try It 1
Write the rule for the nth term for 16, 12, 8, 4, 0, ….
Answer
\(a_n = -4n + 20\)
Example 2: Find Explicit Rule Based on Two Terms
The 7th term of an arithmetic sequence is 26, and the 15th term is 50. Write the rule for the nth term.
Solution
This gives two points \(\color{blue}{a_7 = 26}\) and \(\color{red}{a_{15} = 50}\). Substitute each point into the formula to obtain two equations to solve for a1 and d.
The 5th term of an arithmetic sequence is 24, and the 9th term is 44. Write the rule for the nth term.
Answer
\(a_n = 5n - 1\)
Example 3: Write the Recursive Rule for an Arithmetic Sequence
Write the recursive rule for 4, 1, −2, −5, ….
Solution
The common difference appears to be −3, and the first term is 4. \(d = -3, a_1 = 4\)
$$ a_1 = a_1, a_{n} = a_{n-1} + d $$
$$ a_1 = 4, a_{n} = a_{n-1} - 3 $$
Try It 3
Write the recursive rule for 5, 7, 9, 11, ….
Answer
\(a_1 = 5, a_{n} = a_{n-1} + 2\)
Arithmetic Series
What is the sum of all the odd numbers less than 20? Start listing the numbers.
$$ 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 $$
Bend the last half of the list under the first and add.
1
+
3
+
5
+
7
+
9
+ 19
+
17
+
15
+
13
+
11
20
+
20
+
20
+
20
+
20
5(20) = 100
Notice the first and last term added to 20. Also, notice that there were 10 terms, but the 20 was multiplied by 5 or \(\frac{10}{2}\). Several more tests could be done to get a pattern. It produces the formula for the sum of an arithmetic series.
$$ S_n = \frac{n}{2}\left(a_1 + a_n\right) $$
Sum of an Arithmetic Series
$$ S_n = \frac{n}{2}\left(a_1 + a_n\right) $$
where \(a_n = a_1 + (n-1)d\) and the lower limit is 1.
Example 4: Find the Sum of an Arithmetic Series
Find the sum of the integers 1 to 35.
Solution
This is an arithmetic series with d = 1 and an = 1. The last term is \(a_{35} = 35\), so n = 35.
$$ S_n = \frac{n}{2}\left(a_1 + a_n\right) $$
$$ S_{35} = \frac{35}{2}\left(1 + 35\right) $$
$$ S_{35} = \frac{35}{2} (36) $$
$$ S_{35} = 630 $$
Example 5: Find the Sum of an Arithmetic Series
Find the 40th partial sum of –10 + –7 + –4 + –1 + ….
Solution
$$ S_n = \frac{n}{2} \left(a_1 + a_n\right) $$
From the series, d = 3 and \(a_1 = -10\). an is not known, so use the explicit rule formula for the arithmetic sequence.
$$ a_n = a_1 + (n-1)d $$
$$ a_{n} = -10 + (n-1)(3) $$
$$ a_n = 3n - 13 $$
Because the problem asks for the 40th partial sum, n = 40.
$$ a_{40} = 3(40) - 13 = 107 $$
Now that all the variables are known, fill in the sum formula.
$$ S_n = \frac{n}{2} \left(a_1 + a_n\right) $$
$$ S_{40} = \frac{40}{2}\left(-10 + 107\right) $$
$$ S_{40} = 1940 $$
Try It 4
Find the sum of the first 20 terms of 8 + 7 + 6 + 5 + ….
The explicit rule in this problem is 4i – 20 which is linear, so the problem is arithmetic. n = 80 because that is the upper limit. To find a1 and an plug the 1 and 80 into the \(4i - 20\).
You are saving money to buy your first car. You save $100 every month with a goal of $3000. (a) Write a rule for the nth term for the amount of money you have saved. (b) How many months until you have saved your $3000? (c) And what kind of car do you want?