Chapter 8 Sequences, Series, and the Binomial Theorem

Transcription

1 Chapter 8 Sequences, Series, and the Binomial Theorem Section 1 Section 2 Section 3 Section 4 Sequences and Series Arithmetic Sequences and Partial Sums Geometric Sequences and Series The Binomial Theorem Vocabulary Infinite sequence Recursively Summation (Sigma) notation Arithmetic sequence Geometric sequence Binomial coefficients Finite sequence Factorial Series Finite arithmetic sequence (Infinite) Geometric series The Binomial Theorem Pascal s Triangle Page 132

2 Section 8.1 Sequences and Series Objective: In this lesson you learned how to use sequence, factorial, and summation notation to write the terms and sums of sequences. Important Vocabulary Infinite Sequence Finite Sequence Recursively Factorial Summation (Sigma) Notation Series Arithmetic Sequence Finite Arithmetic Sequence I. Sequences An infinite sequence is: How to use sequence notation to write the terms sequences The function values a 1, a 2, a 3, a 4,, a n, are the of an infinite sequence. A finite sequence is: To find the first three terms of a sequence, given an expression for its nth term, you: To define a sequence recursively, you need to be given. All other terms of the sequence are then defined using. II. Factorial Notation If n is a positive integer, n factorial is defined by. Zero factorial is defined as. How to use factorial notation Page 133

3 III. Summation Notation The sum of the first n terms of a sequence is represented by the summation or sigma notation, n a i = i=1 where i is called the, n is the How to use summation notation to write sums, and 1 is the. Properties of Sums: Sums of Powers of Integers: n = n 2 = n 3 = n 4 = n 5 = Page 134

4 IV. Series The sum of the terms of a finite or infinite sequence is called a(n). How to find sums of infinite series Consider the infinite sequence a 1, a 2, a 3,, a n, The sum of the first n terms of the sequence is called a(n) or the of the sequence and is denoted by a 1 + a 2 + a a i + = n i=1 a i. The sum of all terms of the infinite sequence is called a(n) and is denoted by a 1 + a 2 + a a i + = i=1 a i. Page 135

5 Section 8.1 Examples Sequences and Series ( 1 ) Find the first five terms of the sequence given by a n = 5 + 2n( 1) n. ( 2 ) Write an expression for the nth term a n of the given sequence. 2, 5, 10, 17, ( 3 ) A sequence is defined recursively as follows: a 1 = 3, a k = 2a k 1 + 1, where k 2 Write the first five terms of this sequence. ( 4 ) Evaluate the factorial expression n! (n+1)! ( 5 ) Find the following sum: 7 (2 + 3i) i=2 ( 6 ) For the given series find (a) the third partial sum and (b) the sum i i=1 Page 136

6 Section 8.2 Arithmetic Sequences and Partial Sums Objective: In this lesson you learned how to recognize, write, and use arithmetic sequences. Important Vocabulary Arithmetic sequence Finite Arithmetic Sequence I. Arithmetic Sequences The common difference of an arithmetic sequence is: How to recognize, write, and find the nth terms of arithmetic sequences The nth term of an arithmetic sequence has the form, where d is the common difference between consecutive terms of the sequence, and c = a 1 d. An arithmetic sequence a n = dn + c can be thought of as, after a shift of units from. The nth term of an arithmetic sequence has the alternative recursion formula. II. The Sum of a Finite Arithmetic Sequence The sum of a finite arithmetic sequence with n terms is given by. The sum of the first n terms of an infinite sequence is called the. How to find nth partial sums of arithmetic sequences Page 137

7 Section 8.2 Examples Arithmetic Sequences and Partial Sums ( 1 ) Determine whether or not the following sequence is arithmetic. If it is, find the common difference. 7, 3, 1, 5, 9, ( 2 ) Find a formula for the nth term of the arithmetic sequence whose common difference is 2 and whose first term is 7. ( 3 ) Find the sixth term of the arithmetic sequence that begins with 15 and 12. ( 4 ) Find the sum of the first 20 terms of the sequence with nth term a n = 28 5n. Page 138

8 Section 8.3 Geometric Sequences and Series Objective: In this lesson you learned how to recognize, write, and use geometric sequences. Important Vocabulary Geometric sequence (Infinite) geometric series I. Geometric Sequences The common ratio of a geometric sequence is: How to recognize, write, and find the nth terms of geometric sequences The nth term of a geometric sequence has the form, where r is the common ratio of consecutive terms of the sequence. So, every geometric sequence can be written in the following form:. If you know the nth term of a geometric sequence, you can find the (n + 1)th term by. That is, a n+1 =. II. The Sum of a Finite Geometric Sequence The sum of a geometric sequence a 1, a 1 r, a 1 r 2, a 1 r 3, a 1 r 4,, a 1 r n 1 with common ratio r 1 is given by. How to find nth partial sums of geometric sequences When using the formula for the sum of a geometric sequence, be careful to check that the index begins with i = 1. If the index begins at i = 0: III. Geometric Series If r < 1, then the infinite geometric series a 1 + a 1 r + a 1 r 2 + a 1 r 3 + a 1 r a 1 r n 1 + has the sum. If r > 1, the series a sum. How to find sums of infinite geometric series Page 139

9 Section 8.3 Examples Geometric Sequences and Series ( 1 ) Determine whether or not the following sequence is geometric. If it is, find the common ratio. 60, 30, 0, 30, 60, ( 2 ) Write the first five terms of the geometric sequence whose first term is a 1 = 5 and whose common ratio is 3. ( 3 ) Find the eighth term of the geometric sequence that begins with 15 and 12. ( 4 ) Find the sum: 10 2(0.5) i i=1 ( 5 ) Find the sum: 12 4(0.3) i i=0 ( 6 ) If possible, find the sum: 9(0.25) i 1 i=1 Page 140

10 Section 8.4 The Binomial Theorem Objective: In this lesson you learned how to use the Binomial Theorem and Pascal s Triangle to calculate binomial coefficients and write binomial expansions. Important Vocabulary Binomial coefficients The Binomial Theorem Pascal s Triangle I. Binomial Coefficients List four general observations about the expansion of (x + y) n for various values of n. 1. How to use the Binomial Theorem to calculate binomial coefficients The Binomial Theorem states that in the expansion of (x + y) n = x n + nx n 1 y + + n C r x n r y r + + nxy n 1 + y n, the coefficient of x n r y r is: II. III. Binomial Expansion Writing out the coefficients for a binomial that is raised to a power is called. Pascal s Triangle Construct rows 0 through 6 of Pascal s Triangle. How to use binomial coefficients to write binomial expansions How to use Pascal s Triangle to calculate binomial coefficients Page 141

11 Section 8.4 Examples The Binomial Theorem ( 1 ) Find the binomial coefficient 12 C 5. ( 2 ) Write the expansion of the expression (x + 2) 5 Page 142