Chapter 13: Sequences and Series

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Presentation transcript:

Chapter 13: Sequences and Series Section 13.1: Arithmetic and Geometric Sequences

arithmetic sequence – a sequence of numbers such that the difference of any two consecutive terms is constant. tn = t1 + (n – 1)d To get the start with and add the nth term the first difference term n – 1 times

geometric sequence – a sequence such that the ratio (common ratio) of any two consecutive terms is constant. tn = t1 · r (n – 1) To get the start with and multiply nth term the first by the ratio term n – 1 times

sequence – defined to be a function whose domain is the set of positive integers.

Example 1: State whether the given sequence is arithmetic, geometric, or neither. 1 · 2, 2 · 3, 3 · 4, 4 · 5, … Neither the difference of consecutive terms nor the ratio of consecutive terms is constant. Neither

51/3 , 101/3 , 201/3, 401/3, … The common ratio of the terms is 21/3. geometric 15, 11, 7, 3, … The common difference is -4. Arithmetic

Example 2: State the first four terms of the specified sequence. Then tell whether the sequence is arithmetic, geometric, or neither. tn = 5n + 2 = 5(1) + 2 = 5(2) + 2 = 5(3) + 2 = 5(4) + 2 7, 12, 17, 22, … arithmetic

tn = n + 1 n + 2 tn = 3n tn = n3 b) 2/3, ¾, 4/5, 5/6, … (neither) 3, 9, 27, 81, … (geometric) 1, 8, 27, 64, …(neither)

Example 3: Find a formula for tn. 1, 4, 7, 10, … b) 8, 4, 2, 1, …

HOMEWORK (Day 1) pg. 476; 1 – 8, 13, 15

Example 4: a. Arithmetic sequence t1 = 114, t3 = 100, Find t20 Example 4: a. Arithmetic sequence t1 = 114, t3 = 100, Find t20. 100 = 114 + (3 – 1)d -14 = 2d -7 = d t20 = 114 + (20 – 1)(-7) t20 = -19

Geometric sequence t1 = 5/64, t4 = 5/8, find t26. 5/8 = 5/64(r)(4 – 1) 8 = r3 r = 2 t26 = 5/64(2)(26-1) t26 = 2,621,440

Practice Arithmetic Sequence t1 = 5, t7 = 29, find t53. d = 4 Geometric Sequence t1 = ½ , t5 = 8, find t10. r = 2 t10 = 256

HOMEWORK (Day 2) pg. 477; 29, 30, 33, 34, 36

Example 5: In a certain sequence, t2 = 2 and t5 = 16. Find t10 if the sequence is: Arithmetic sequence t2 = 2 2 = t1 + (2 – 1)d 2 = t1 + d

t5 = 16 16 = t1 + (5 – 1)d 16 = t1 + 4d (2 = t1 + d) 14 = 3d d = 14/3

2 = t1 + d t1 = 2 – d t1 = 2 – 14/3 t1 = -8/3 t10 = t1 + (10 – 1)d = -8/3 + (9)(14/3) = 118 3

Geometric sequence t2 = 2 2 = t1 · r2 – 1 2 = t1 · r t5 = 16 16 = t1 · r5 – 1 16 = t1 · r4 t1 = 2 r t1 = 16 r4

2 = 16 r r4 2r4 = 16r r3 = 8 r = 2 t1 = 2/r = 2/2 = 1 t10 = t1 · r10 – 1 = 1 · r10 – 1 = 1 · 29 = 512

HOMEWORK (Day 3) pg. 477; 31, 32, 35

Arithmetic; t1 = 3, t5 = 23, find t20. Geometric; t1 = 24, t6 = ¾ , find t10. Arithmetic; t2 = 5, t8 = 23, find t30. Geometric; t4 = 32, t10 = 2,048, find t25.