Examples of Finite Sequences (finite terms) Examples of Infinite Sequences (infinite terms)

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1 Math 120 Intermediate Algebra Sec 10.1: Sequences Defn A sequence is a function whose domain is the set of positive integers. The formula for the nth term of a sequence is called the general term. Examples of Finite Sequences (finite terms) {f(1), f(2), f(3), f(4), f(5), f(6)} where f(n) = 2n { } {f(1), f(2), f(3), f(4), f(5), f(6)} where f(n) = 1/n { } Instead of writing f(n) = n 2, we would write a n = n 2. List the first 4 terms of {a n }. { 3, 6, 9, 12, 15,, 30} a 1 Examples of Infinite Sequences (infinite terms) {b n } = {( 2 3 ) n} b n = {a n } = { ( 1)n n n + 2 } { ( 1)n+1 } n Ex 1 #22 Write the first five terms of the sequence. { 3n 3 n} Ex 2 Write the nth term of each sequence suggested by the pattern. a) #26 b) #28 c) #30 d) #32 5, 10, 15, 20, 1 2, 1, 3 2, 2, 5 2, 0, 7, 26, 63, 1, 1 2, 1 4, 1 8, Page 1 of 8

2 Defn The sum of the first n terms of a sequence, a 1 + a 2 + a a n, is expressed using summation notation as follows: n a i = a 1 + a 2 + a a n i=1 The symbol Σ is the instruction to sum, or add up, the terms. The integer i is called the index of the sum; it tells us where to start and end the sum. When there are a finite number of term to be added, we call it a partial sum. Ex 3 Write out each sum and determine its value. a) #40 b) # [( 1) k k] 2 k=1 j=1 Ex 4 Express each sum using summation notation. a) #48 b) # ( 1)15+1 ( 2 3 ) 15 Ex 5 #57 Let u n = (1 + 5)n (1 5) n 2 n 5 define the nth term of a sequence. Find the first 10 terms of the sequence. This sequence is called the sequence. The terms of the sequence are called the Fibonacci numbers. Page 2 of 8

3 Sec 10.2: Arithmetic Sequences Defn When the difference between successive terms in a sequence is constant (always the same number), the sequence is called an arithmetic sequence (or arithmetic progression). Call the first term a 1 and the common difference d, the terms of the arithmetic sequence follow the pattern: a 1, a 1 + d, a 1 + 2d, a 1 + 3d, Ex 6 #20 Verify that the given sequence is arithmetic. Then find the first term and the common difference. {5 2n} Formula for nth Term of an Arithmetic Sequence For the arithmetic sequence {a n } whose first term is a 1 and whose common difference is d, the nth term is determined by the formula a n = a 1 + (n 1)d Ex 7 #28 Find a formula for the nth term of the arithmetic sequence. a 1 = 3; d = 1 2 Sum of the First n Terms of an Arithmetic Sequence Let {a n } be an arithmetic sequence with first term a 1 and common difference d. The sum S n of the first n terms of {a n } is S n = n 2 {2a 1 + (n 1)d] or S n = n 2 (a 1 + a n ) Ex 8 Write a formula for the nth term of each arithmetic sequence. Use the formula to find the 20 th term in each sequence. a) #34 b) #36 20, 14, 8, 2, 10, 19 2, 9, 17 2, Page 3 of 8

4 Ex 9 #44 Find the first term and the common difference of the arithmetic sequence described. Give a formula for the nth term of the sequence. 5 th term is 5; 13 th term is 7 Ex 10 #55 Find the sum of the first 30 terms of the arithmetic sequence { n}. Sec 10.3: Geometric Sequences and Series Defn If the ratio of consecutive terms in a sequence is constant, then the sequence is a geometric sequence. Let the first term be a 1 and the common ratio of consecutive terms be r. Then the terms of the geometric sequence follow the pattern: a 1, a 1 r, a 1 r 2, a 1 r 3, Examples of Geometric Sequences {3, 12, 16, 64, } {2, 6, 18, 48, } { 1 2, 1 4, 1 8, 1 16, 1 32, } Finding the nth Term of a Geometric Sequence If a geometric sequence {a n } has first term a 1 and common ratio r, then the nth term is determined by the formula a n = a 1 r n 1 r 0. Page 4 of 8

5 Ex 11 #26 common ratio. { 3 n 2 n 1} Show that the given sequence is a geometric sequence. Then find the first term and Ex 12 #32 Find a formula for the nth term of the geometric sequence. Next, use the formula to find the 8 th term. a 1 = 1; r = 4 Ex 13 Find the indicated term of each geometric sequence. a) #37 b) #39 15 th term of 4, 2, 1, 1, 2 9th term of 0.5, 0.05, 0.55, , Sum of the First n Terms of a Geometric Sequence (Partial Sum of a Geometric Series) Let {a n } be a geometric sequence with first term a 1 and common ratio r, where r 0,1. The sum S n of the first n terms of {a n } is S n = a 1 1 rn 1 r, r 0, 1 Ex 14 #44 Find the sum. If necessary, express your answer to as many decimal places as your calculator allows (1 2 ) Page 5 of 8

6 Sum of a Geometric Series An infinite sum of the form a 1 + a 1 r + a 1 r a 1 r n 1 + whose first term is a 1 and whose common ratio is r, is called an infinite geometric series and is denoted by a 1 r n 1 n=1 If 1 < r < 1, the sum of the terms of an infinite geometric series with first term a 1 and common ratio r is a 1 r n 1 n=1 = a 1 1 r Ex 15 #58 (100 ( 1 2 ) n 1) n=1 Find the sum. Ex 16 #62 Express the repeating decimal as a fraction in lowest terms. Ex 17 #82 A ball is dropped from a height of 30 feet. Each time it strikes the ground, it bounces up to 0.8 of the previous height. a) What height will the ball bounce up to after it strikes the ground for the 4 th time? b) How high will it bounce after it strikes the ground for the 5 th time? Page 6 of 8

7 c) How many times does the ball need to strike the ground before its bounce is less than 7 inches? d) What total distance does the ball travel before it stops bouncing? Sec 10.4: The Binomial Theorem Defn If n 0 is an integer, the factorial symbol n! (read n factorial ) is defined as 0! = 1 1! = 1 n! = n(n 1)(n 2) if n 2 Ex 18 Evaluate each factorial expression. a) b) #18 5! 9! 6! Defn j ) is defined as If j and n are integers with 0 j n, the symbol ( n ) (read n taken j at a time or n choose j ( n j ) = n! j! (n j)! Useful formulas: ( n 0 ) = 1 (n n ) = n ( 1 n 1 ) = n (n n ) = 1 Ex 19 #22 ( 12 9 ) Evaluate. Page 7 of 8

8 Pascal s Triangle ( 0 0 ) ( 1 0 ) (1 1 ) ( 2 0 ) (2 1 ) (2 2 ) ( 3 0 ) (3 1 ) (3 2 ) (3 3 ) ( 4 0 ) (4 1 ) (4 2 ) (4 3 ) (4 4 ) ( 5 0 ) (5 1 ) (5 2 ) (5 3 ) (5 4 ) (5 5 ) The Binomial Theorem Let x and a be real numbers. For any positive integer n, (x + a) n = ( n 0 ) xn + ( n 1 ) axn 1 + ( n 2 ) a2 x n ( n j ) aj x n j + + ( n n ) an The number ( n ) is called a binomial coefficient. j Ex 20 Expand each expression using the Binomial Theorem. a) #26 (x + 5) 5 b) (x 3) 4 c) #30 (3w 4) 4 Page 8 of 8