Arithmetic and Geometric Sequences: Properties and Formulas, Lecture notes of Calculus

Download Arithmetic and Geometric Sequences: Properties and Formulas and more Lecture notes Calculus in PDF only on Docsity! (Chapter 9: Discrete Math) 9.11 SECTION 9.2: ARITHMETIC SEQUENCES and PARTIAL SUMS PART A: WHAT IS AN ARITHMETIC SEQUENCE? The following appears to be an example of an arithmetic (stress on the “me”) sequence: a 1 = 2 a 2 = 5 a 3 = 8 a 4 = 11  We begin with 2. After that, we successively add 3 to obtain the other terms of the sequence. An arithmetic sequence is determined by: • Its initial term Here, it is a1 , although, in other examples, it could be a0 or something else. Here, a1 = 2 . • Its common difference This is denoted by d . It is the number that is always added to a previous term to obtain the following term. Here, d = 3. Observe that: d = a 2 − a 1 = a 3 − a 2 = … = a k+1 − a k k ∈Z+( ) = … The following information completely determines our sequence: The sequence is arithmetic. (Initial term) a1 = 2 (Common difference) d = 3 (Chapter 9: Discrete Math) 9.12 In general, a recursive definition for an arithmetic sequence that begins with a1 may be given by: a 1 given a k+1 = a k + d k ≥ 1; "k is an integer" is implied( ) ⎧ ⎨ ⎪ ⎩⎪ Example The arithmetic sequence 25, 20, 15, 10, … can be described by: a 1 = 25 d = −5 ⎧ ⎨ ⎪ ⎩⎪ (Chapter 9: Discrete Math) 9.15 SECTION 9.3: GEOMETRIC SEQUENCES, PARTIAL SUMS, and SERIES PART A: WHAT IS A GEOMETRIC SEQUENCE? The following appears to be an example of a geometric sequence: a 1 = 2 a 2 = 6 a 3 = 18 a 4 = 54  We begin with 2. After that, we successively multiply by 3 to obtain the other terms of the sequence. Recall that, for an arithmetic sequence, we successively add. A geometric sequence is determined by: • Its initial term Here, it is a1 , although, in other examples, it could be a0 or something else. Here, a1 = 2 . • Its common ratio This is denoted by r. It is the number that we always multiply the previous term by to obtain the following term. Here, r = 3. Observe that: r = a 2 a 1 = a 3 a 2 = … = a k+1 a k k ∈Z+( ) = … The following information completely determines our sequence: The sequence is geometric. (Initial term) a1 = 2 (Common ratio) r = 3 (Chapter 9: Discrete Math) 9.16 In general, a recursive definition for a geometric sequence that begins with a1 may be given by: a 1 given a k+1 = a k ⋅ r k ≥ 1; "k is an integer" is implied( ) ⎧ ⎨ ⎪ ⎩⎪ We assume a1 ≠ 0 and r ≠ 0 . Example The geometric sequence 2, 6, 18, 54, … can be described by: a 1 = 2 r = 3 ⎧ ⎨ ⎪ ⎩⎪ (Chapter 9: Discrete Math) 9.17 PART B : FORMULA FOR THE GENERAL nth TERM OF A GEOMETRIC SEQUENCE Let’s begin with a1 and keep multiplying by r until we obtain an expression for an , where n ∈Z+ . a 1 = a 1 a 2 = a 1 ⋅ r a 3 = a 1 ⋅ r 2 a 4 = a 1 ⋅ r3  a n = a 1 ⋅ r n−1 The general n th term of a geometric sequence with initial term a1 and common ratio r is given by: an = a 1 ⋅ r n−1 Think: As with arithmetic sequences, we take n −1 steps to get from a1 to an . Note: Observe that the expression for an is exponential in n. This reflects the fact that geometric sequences often arise from exponential models, for example those involving compound interest or population growth. (Chapter 9: Discrete Math) 9.20 Example Find the 6th partial sum of the geometric sequence 2, −1, 1 2 , … Solution Recall that a1 = 2 and r = − 1 2 for this sequence. We found in the previous Example that: a 6 = − 1 16 We will use our formula to evaluate: S 6 = 2 − 1 + 1 2 − 1 4 + 1 8 − 1 16 Using our formula directly: S n = a 1 − a 1 r n 1− r or a 1 1− r n 1− r ⎛ ⎝⎜ ⎞ ⎠⎟ If we use the second version on the right … S n = a 1 1− r n 1− r ⎛ ⎝⎜ ⎞ ⎠⎟ S 6 = 2 1− − 1 2 ⎛ ⎝⎜ ⎞ ⎠⎟ 6 1− − 1 2 ⎛ ⎝⎜ ⎞ ⎠⎟ ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ = 2 1− 1 64 1+ 1 2 ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ (Chapter 9: Discrete Math) 9.21 = 2 63 64 3 2 ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ = 2 21 63 32 64 ⋅ 2 1 31 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ = 2 21 32 16 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = 21 16 We can also use the first version and the “first in – first out” idea: S 6 = 2 − 1 + 1 2 − 1 4 + 1 8 − 1 16 “First out” is: a 7 = 1 32 S n = a 1 − a 1 r n 1− r S 6 = 2 − 1 32 1− − 1 2 ⎛ ⎝⎜ ⎞ ⎠⎟ = 63 32 3 2 ← 64 32 − 1 32 ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ = 21 63 16 32 ⋅ 2 1 31 = 21 16 (Chapter 9: Discrete Math) 9.22 PART D: INFINITE GEOMETRIC SERIES An infinite series converges (i.e., has a sum) ⇔ The Sn partial sums approach a real number as n→∞( ) , which is then called the sum of the series. In other words, if lim n→∞ S n = S , where S is a real number, then S is the sum of the series. Example Consider the geometric series: 1 2 + 1 4 + 1 8 + 1 16 + ... Let’s take a look at the partial (cumulative) sums: 1 2 S1= 1 2  + 1 4 S2= 3 4    + 1 8 S3= 7 8    + 1 16 S4= 15 16    + ... It appears that the partial sums are approaching 1. In fact, they are; we will have a formula for this. This series has a sum, and it is 1. The figure below may make you a believer: