Precalculus by Richard Wright

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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. $$1, 1+2, 1+2+2, 1+2+2+2, 1+2+2+2+2$$ $$1, 1+\color{blue}{1}(2), 1+\color{blue}{2}(2), 1+\color{blue}{3}(2), 1+\color{blue}{4}(2)$$ The blue terms are 1 less than the term number, so the rule becomes $$a_n = 1 + (n - 1)2$$ Notice that the first term was 1 and the common difference was 2, replace those numbers in the rule to get the general formula for the nth term. $$a_n = a_1 + (n - 1)d$$ ###### Explicit Rule for Arithmetic Sequences $$a_n = a_1 + (n - 1)d$$ where a1 is the first term and d is the common difference This simplifies to $$a_n = dn + c$$ where $$c = a_1 - d$$. This is linear, so any time an explicit rule for a sequence is linear, it is arithmetic. ###### Recursive Rule for Arithmetic Sequence $$a_1 = a_1, a_{n} = a_{n-1} + d$$ #### Example 1: Write an Explicit Rule for Arithmetic Sequence Write the rule for the nth term for 4, 7, 10, 13, 16, …. ###### Solution The first term is 4, so $$a_1 = 4$$. Subtract terms to find the common difference, d. $$7 - 4 = 3; 10 - 7 = 3; 13 - 10 = 3; 16 - 13 = 3$$ 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. $$a_n = a_1 + (n - 1)d$$ \begin{align} \color{blue}{26} &= \color{blue}{a_1 + (7 - 1)d} \\ \color{red}{50} &= \color{red}{a_1 + (15 - 1)d} \end{align} \begin{align} \color{blue}{26} &= \color{blue}{a_1 + 6d} \\ \color{red}{50} &= \color{red}{a_1 + 14d} \end{align} Solve the system of equations with something like elimination (or substitution or matrices or Cramer's Rule). \require{enclose} \begin{align} (-1)\color{blue}{26} &= \color{blue}{a_1 + 6d} \\ \color{red}{50} &= \color{red}{a_1 + 14d} \\ \enclose{top}{24} &= \enclose{top}{\quad \quad 8d} \end{align} $$d = 3$$ $$26 = a_1 + 6d$$ $$26 = a_1 + 6(3)$$ $$a_1 = 8$$ Now write the rule for the nth term. $$a_n = a_1 + (n - 1)d$$ $$a_n = 8 + (n - 1)3$$ $$a_n = 8 + 3n -3$$ $$a_n = 3n + 5$$ ##### Try It 2 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 + …. ###### Answer –30 #### Example 6: Find the Sum of an Arithmetic Series Evaluate $$\displaystyle \sum_{i=1}^{80} (4i - 20)$$. ###### Solution 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 $$4i - 20$$. $$a_1 = 4(1) - 20 = -16$$ $$a_{80} = 4(80) - 20 = 300$$ $$S_{n} = \frac{n}{2} \left(a_1 + a_n\right)$$ $$S_{80} = \frac{80}{2} \left(a_1 + a_{80}\right)$$ $$S_{80} = \frac{80}{2} (-16 + 300)$$ $$S_{80} = 11360$$ ##### Try It 5 Evaluate $$\displaystyle \sum_{i=1}^{100} (-2i + 16)$$ ###### Answer –8500 ##### Lesson Summary ###### Explicit Rule for Arithmetic Sequences $$a_n = a_1 + (n - 1)d$$ where a1 is the first term and d is the common difference This simplifies to $$a_n = dn + c$$ where $$c = a_1 - d$$. This is linear, so any time an explicit rule for a sequence is linear, it is arithmetic. ###### Recursive Rule for Arithmetic Sequence $$a_1 = a_1, a_{n} = a_{n-1} + d$$ ###### 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. Helpful videos about this lesson. ## Practice Exercises Write the rule for the nth term. 1. 12, 14, 16, 18, 20, … 2. 15, 20, 25, 30, 35, … 3. –4, –1, 2, 5, 8, … 4. The 6th term of an arithmetic sequence is 63, and the 10th term is 107. 5. The 12th term of an arithmetic sequence is –80, and the 20th term is –136. 6. Write the recursive rule for the sequence. 7. 4, 10, 16, 22, 28, … 8. 25, 12, –1, –14, –27, … 9. –50, –29, –8, 13, 34, … 10. Find the sum of the series. 11. $$16 + 15 + 14 + 13 + \cdots + -4$$ 12. Find the 13th partial sum: $$53 + 57 + 61 + 65 + 69 + \cdots$$ 13. Find the 100th partial sum: $$-34 + -36 + -38 + -40 + -42 + \cdots$$ 14. $$\displaystyle \sum_{i=1}^{15} (8i - 50)$$ 15. $$\displaystyle \sum_{n=1}^{20} (-4n + 3)$$ 16. $$\displaystyle \sum_{k=1}^{50} (12k - 1)$$ 17. Problem Solving 18. You are saving money to buy your first car. You save100 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?
19. Mixed Review

20. (10-02) Evaluate $$\displaystyle \sum_{i=4}^{8} (2i + 10)$$.
21. (10-02) Evaluate $$\displaystyle \sum_{n=1}^{25} 2n^2$$.
22. (10-01) Write the first five terms of $$a_n = n^2 - n$$.
23. (9-06) Use Cramer's Rule to solve \left\{\begin{align} 2x - 3y &= -4 \\ 4x + 5y &= 14 \end{align}\right..
24. (9-04) Use an inverse matrix to solve \left\{\begin{align} 3x - 3y &= 0 \\ 2x - y &= 2 \end{align}\right..

1. $$a_n = 2n + 10$$
2. $$a_n = 5n + 10$$
3. $$a_n = 3n - 7$$
4. $$a_n = 11n - 3$$
5. $$a_n = -7n + 4$$
6. $$a_1 = 4, a_{n} = a_{n-1} + 6$$
7. $$a_1 = 25, a_{n} = a_{n-1} - 13$$
8. $$a_1 = -50, a_{n} = a_{n-1} +21$$
9. 126
10. 1001
11. –13300
12. 210
13. –780
14. 15250
15. an = 100n; 30 months
16. 110
17. 11050
18. 0, 2, 6, 12, 20
19. (1, 2)
20. (2, 2)