University:
Davenport University
Course:
MATH 120 | College Mathematics
Academic year:

2022
Views:

319

Pages:

3
Author:

Kronichjfv

Arithmetic and Geometric Sequences A sequence is a list of numbers or objects, called terms, in a certain order. In an arithmetic sequence, the difference between one term and the next is always the same. This difference is called a common difference. The common difference is added to each term to get the next term. 2, 5, 8, 11, 14, … This is an increasing arithmetic sequence with a common difference of 3. 32, 26, 20, 14, 8, … This is a decreasing arithmetic sequence with a common difference of –6. Example: What are the next three terms in the sequence? 1, 5, 9, 13, … I can see that this is an arithmetic sequence with a common difference of 4. To get the next three terms, add 4 to 13 which equals 17, the next term in the sequence. Then add 4 to 17 to get the next term to get 21, etc. So the next three terms are 17, 21, and 25. Use the following formula to find any term of an arithmetic sequence. an = a1 + ( n − 1) d an = the term in the sequence you are trying to find (n represents the desired term number) a1 = the first term in the sequence d = the common difference Example: What is the 10th term of the following sequence? 1, 5, 9, 13, … a10 = 1 + (10 − 1)4 = 1 + 9 ⋅ 4 = 1 + 36 = 37 So the 10th term of this sequence is 37. Example: What is the 12th term of the following sequence? 34, 31, 28, 25, 22, … a12 = 34 + (12 − 1)( −3) = 34 + 11( −3) = 34 + ( −33) = 1 The 12th term of this sequence is 1. A geometric sequence is a sequence of numbers where the ratio of consecutive terms is constant. This ratio is called the common ratio (r). Sometimes the terms of a geometric sequence get so large that you may need to express the terms in scientific notation rounded to the nearest tenth. 2, 6, 18, 54, … This is an increasing geometric sequence with a common ratio of 3. 1,000, 200, 40, 8, … This is a decreasing geometric sequence with a common ratio or 0.2 or 1 . 5 Example: What are the next three terms of the following sequence? 4, 20, 100, 500, … 500 ⋅ 5 = 2500 2500 ⋅ 5 = 12,500 The next three terms are 2,500, 12,500, and 62,500. 12,500 ⋅ 5 = 62,500 Explicit sequences also have a formula for finding any term in a sequence. an = a1 r ( n −1) an = the term in the sequence you are trying to find (n represents the desired term number) a1 = the first term in the sequence r = the common ratio Example: Find the 7th term in the following sequence: 6, 18, 54, 162, … Finding the common ratio can be harder than finding the common difference. One way to find it is the divide each term by the term before it. 18 ÷ 6 = 3 , 54 ÷ 18 = 3 , 162 ÷ 54 = 3 So the common ratio is 3. −1) = a7 6 ⋅ 3(7 = 6= ⋅ 36 6 ⋅ 729 = 4, 374 So the 7th term of the sequence is 4, 374. Example: Find the 8th term in the following sequence: 96, 48, 24, 12, 6, … To find the common ratio, divide each term by the one before it. 48 ÷ 96 = 1 1 1 1 , 24 ÷ 48 = , 12 ÷ 24 = The common ratio is . 2 2 2 2 = a8 96 ⋅ 1 (8 −1) = 2 1 7 96 = ⋅ 2 1 96= ⋅ 128 0.75 The 8th term of the sequence is 0.75.