Arithmetic
| Site: | Clare-Gladwin RESD |
| Course: | Michigan Algebra II KHauck |
| Book: | Arithmetic |
| Printed by: | Guest user |
| Date: | Sunday, May 19, 2024, 7:21 PM |
Description
Arithmetic
An arithmetic sequence is a sequence that adds a constant from one term to the next. The common value added or subtracted between two successive terms of an arithmetic sequence is called the common difference, d.
An example of an arithmetic sequence is 4, 7, 10, 13, 16 … and the common difference of this sequence is 3. To change one term to the next in the sequence, add three.
In general, the terms of a sequence are written as: a1, a2, a3, a4 ...anwhere a1 is the first term in the sequence, a2is the second term in the sequence, a3 is the third term in the sequence, and an is the nth term in the sequence. It is called the nth term because sequences will have a varying number of terms.
Example 1
Find the common difference and the next term of the following sequence:
3, 11, 19, 27, 35...
Step 1. To find the common difference, subtract any consecutive pair of terms.
The difference is always 8, so d = 8.
Step 2. Use the common difference to find the next term.
Since the last term was 35, adding the common difference of 8 will find the next term. The next term is 35 + 8 or 43.
Example 2
Write the first four terms of a sequence given that the nth term is an = 2n + 3.
Step 1. Find the first term. To find the first term, a1, substitute n = 1 into the formula and solve.
a1 = 2(1) + 3
a1 = 5
Step 2. Find the second term. To find the second term, a2, substitute n = 2 into the formula and solve.
a2 = 2(2) + 3
a2 = 7
Step3. Find the third term.
a3 = 2(3) + 3
a3 = 9
Step 4. Find the fourth term.
a4 = 2(4) + 3
a4 = 11
The first four terms of the sequence are 5, 7, 9, and 11.
Explicit Models
The following steps lead to an important discovery. Remember, when finding the next term in an arithmetic sequence, add the common difference, d.
Second Term: a 2 = a1 + d
Third Term: a 3 = a2 + d = (a1 + d) + d = a1 + 2 d
Nth Term: an = a1 + (n - 1)d
The formula above is also known as the explicit formula and is used to determine the nth term when the previous term is unknown. When simplified, this formula becomes: an = dn + (a1 - d) and is linear. The common difference is the slope and (a1 - d) is the y-intercept.
Example 1
Find the first four terms of an arithmetic sequence whose first term is -1 and whose difference is 4.
Step 1. Use the first term and the common difference to find the terms.
-1 + 4 = 3
3 + 4 = 7
7 + 4 = 11
Therefore, the first four terms are -1, 3, 7, and 11.
Example 2
Find the 15th term of an arithmetic sequence whose first term is -1 and whose common difference is 4.
Step 1. Use the formula to determine the explicit model.
an = a1+ d(n - 1)
an = -1 + 4(n - 1)
Step 2. Substitute the position of the term into the model and solve; n = 15.
a15 = -1 + 4(15 - 1)
a15 = -1 + 4(14)
a15 = 55
Example 3
Find the explicit model for the following sequence:
5, 9, 13, 17, 21 …
Step 1. Determine the common difference.
9 - 5 = 4
13 - 9 = 4
Every difference is four, d = 4.
Step 2. Substitute d = 4 and a = 5 into the formula and simplify.
an = 5 + 4(n - 1)
an = 5 + 4n - 4
an = 4n + 1
Example 4
Find the 39th term of an arithmetic sequence whose first term is -3 and whose 24th term is 89.
Step 1. Substitute the values into the formula and solve for d.
an = a1+(n-1)d
a24 = -3+(24-1) d92 = (23) d
89 = -3+(23) d
d = 4Step 2. Substitute d = 4 and a = -3 and n = 39 into the formula and solve.
an = a1 + d(n - 1)
a39 = -3 + 4(39 - 1)
a39 = -3 + 4(38)
a39 = -3 + 152
a39 = 149
Recursive Models
Another way to represent an arithmetic sequence is a recursive function, or any model that demonstrates how to find the next term in a sequence. In order to find the next term, an, add the common difference to the previous term, an-1. An example of a recursive model is: an = an-1 + d. This formula is also known as the recursive formula and is used when previous terms and the common difference are known values.
Example Write a recursive function to model the following sequence:
-3, -1, 1, 3, 5 …
Step 1. Determine the common difference.
-1 - (-3) = 2
1 - (-1) = 2
The common difference is two, d = 2.
Step 2. Write a recursive model and reference the first term.
a1 = -3
an = an -1 + 2
Video Lessons
To learn how to find common differences, select the following link:
Identifying Arithmetic Sequences
To learn how to find the general formula for arithmetic sequences, select the following link:
Finding the nth Term Given an Arithmetic Sequence
To learn how to find the nth term of arithmetic sequences, select the following link:
Finding the nth Term Given Two TermsGuided Practice
To solidify your understanding of arithmetic sequences, visit the following link to Holt, Rinehart, and Winston Homework Help Online. It provides examples, video tutorials and interactive practice with answers available. The Practice and Problem Solving section has two parts. The first part offers practice with a complete video explanation for the type of problem with just a click of the video icon. The second part offers practice with the solution for each problem only a click of the light bulb away.
Guided PracticePractice
Arithmetic Sequences Worksheet
*Note: If Google Docs displays, "Sorry, we were unable to retrieve the document for viewing," refresh your browser.
Answer Key
Arithmetic Sequences Answer Key
*Note: If Google Docs displays, "Sorry, we were unable to retrieve the document for viewing," refresh your browser.
Sources
Sources used in this book:
Embracing Mathematics, Assessment & Technology in High Schools; a Michigan Mathematics & Science Partnership Grant Project Florida Virtual School, http://www.flvs.net/ (accessed 2/25/2010).
Holt, Rinehart & Winston, "Guided Practice." http://go.hrw.com/math/midma/gradecontent/loadlesson.html?course=c3&chapter=12&lesson=1&SE=1&sz_audio=1&calc=1&state=xx&actCourse=4 (accessed 7/13/2010).
Holt, Rinehart & Winston, "Sequences and Series." http://my.hrw.com/math06_07/nsmedia/homework_help/alg2/alg2_ch12_03_homeworkhelp.html (accessed 7/13/2010).
Stapel, Elizabeth. "Arithmetic and Geometric Sequences." http://www.purplemath.com/modules/series3.htm (accessed 2/25/2010).