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5.7

5.7. Arithmetic and Geometric Sequences. 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. Sequences.

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5.7

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  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.

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