1 / 11

Understanding Arithmetic and Geometric Sequences

Learn how to write terms for arithmetic and geometric sequences, including formulas for general terms. Practice examples provided.

vgalvan
Download Presentation

Understanding Arithmetic and Geometric Sequences

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. 5.7 Arithmetic and Geometric Sequences

  2. Objectives • Write terms of an arithmetic sequence. • Use the formula for the general term of an arithmetic sequence. • Write terms of a geometric sequence. • Use the formula for the general term of a geometric sequence.

  3. Sequences • A sequence is a list of numbers that are related to each other by a rule. • The numbers in the sequence are called its terms. For example, a Fibonacci sequence term takes the sum of the two previous successive terms, i.e., 1+1=2 1+2=3 3+2=5 5+3=8

  4. Arithmetic Sequences • An arithmetic sequence is a sequence in which each term after the first differs from the preceding term by a constant amount. • The difference between consecutive terms is called the common difference of the sequence.

  5. Example 1: Writing the Terms of an Arithmetic Sequence Write the first six terms of the arithmetic sequence with first term 6 and common difference 4. Solution: The first term is 6. The second term is 6 + 4 = 10. The third term is 10 + 4 = 14, and so on. The first six terms are 6, 10, 14, 18, 22, and 26

  6. The General Term of an Arithmetic Sequence • Consider an arithmetic sequence with first term a1. Then the first six terms are • Using the pattern of the terms results in the following formula for the general term, or the nth term, of an arithmetic sequence: The nth term (general term) of an arithmetic sequence with first term a1 and common difference d is an = a1 + (n – 1)d.

  7. Example 3: Using the Formula for the General Term of an Arithmetic Sequence Find the eighth term of the arithmetic sequence whose first term is 4 and whose common difference is 7. Solution: To find the eighth term, a8, we replace n in the formula with 8, a1 with 4, and d with 7. an = a1 + (n – 1)d a8 = 4+ (8 – 1)(7) = 4 + 7(7) = 4 + (49) = 45 The eighth term is 45.

  8. Geometric Sequences • A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a fixed nonzero constant. • The amount by which we multiply each time is called the common ratio of the sequence.

  9. Example 5: Writing the Terms of a Geometric Sequences Write the first six terms of the geometric sequence with first term 6 and common ratio ⅓. Solution: The first term is 6. The second term is 6 · ⅓ = 2. The third term is 2 · ⅓ = ⅔, and so on. The first six terms are

  10. The General Term of a Geometric Sequence • Consider a geometric sequence with first term a1 and common ratio r. Then the first six terms are • Using the pattern of the terms results in the following formula for the general term, or the nth term, of a geometric sequence: The nth term (general term) of a geometric sequence with first term a1 and common ratio r is an = a1r n-1

  11. Example 6: Using the Formula for the General Term of a Geometric Sequence Find the eighth term in the geometric sequence whose first term is 4 and whose common ratio is 2. Solution: To find the eighth term, a8, we replace n in the formula with 8, a1 with 4, and r with 2. an = a1r n-1 a8 = 4(2)8-1 = 4(2)7 = 4(128) = 512 The eighth term is 512.

More Related