A sequence is a function that assigns to each positive integer n = 1, 2, 3,, a number a n. S n = a 1 + a 2 + a a n.

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1 6. Sequences, Series and Arithmetic Sequence Summary A sequence is a function that assigns to each positive integer n =, 2, 3,, a number a n. a n is called the n th term of the sequence. The terms a, a 2, a 3,... are called the first, second, third, etc. n is referred to as the index. Generally, a sequence can be written in one of the following forms: a, a 2, a 3,... or {a, a 2, a 3,... } or briefly {a n } A series is the indicated sum of the terms of a sequence. Thus if a, a 2, a 3,... a n,... is a sequence, then associated with the sequence is the series given by S n = a + a 2 + a a n Series are often written using the summation notation: S n = a + a 2 + a a n = Here called the summation symbol and k is called the index of summation. The right hand side of this definition is read, the sum of a k with k going from to n. ARITHMETIC SEQUENCES: A sequence {a n } is called Arithmetic Sequence if the difference of any two successive terms a n and a n is a constant d called common difference. a n a n = d If the first term of an arithmetic sequence a and the common difference d are known, then the n th term is defined by: a n = a + (n )d The sum of the first n terms of an arithmetic sequence is given by: n a k S n = n 2 (a + a n ) = n 2 [2a + (n )d] pg.

2 Example() Write the first four terms for each of the sequences defined below: a. a n = 2n b. a n = 3n c. a n = 3 ( 2) n a. a n = 2n; n = a = 2() = 2 n = 2 a 2 = 2(2) = 4 n = 3 a 3 = 2(3) = 6 n = 4 a 4 = 2(4) = 8 The sequence would be written as: 2, 4, 6, 8,... b. a n = 3n ; n = a = 3() = 2 n = 2 a 2 = 3(2) = 5 n = 3 a 3 = 3(3) = 8 n = 4 a 4 = 3(4) = The sequence would then be written as: 2, 5, 8,,... c. a n = 3 ( 2) n ; n = a = 3( 2) = 6 n = 2 a 2 = 3( 2) 2 = 2 n = 3 a 3 = 3( 2) 3 = 24 n = 4 a 4 = 3( 2) 4 = 48 The sequence would then be written as: 6, 2, 24, 48,... pg. 2

3 Example(2) Find an expression (formula) for the n th term of the following sequences: a. 3, 5, 7,... b. 4, 8, 2, 6,... a. 3, 5, 7,... Comparing the given terms with each other, it s clear that only the denominator changes from one term to the next one and takes the odd numbers starting with 3 for n =. Odd numbers can be generated from the formula (2n + ). Therefore, the n th term of this sequence is: b. 4, 8, 2, 6,... a n = 2n + This sequence is clearly formed from terms that are multiples of the number 4: 4, 4 2, 4 3,... Therefore, the n th is: a n = 4n Example(3) Write each series in expanded form. a. 4 (6k + ) b. 5 k 2 c. 20 k=3 ( ) k (5k) a. Replace k, in turn, with the integers from to 4 and add the result 4 (6k + ) = (6 + ) + (6 2 + ) + (6 3 + ) + (6 4 + ) b. Following the same steps as in (a) above, we get = = 64. k 2 = 3 2 pg. 3

4 c. Note that the index k starts from 3; there is no requirement that a series must start from index, so as in (a) and (b) above, we have that 20 k=3 ( ) k (5k) = Also note that if there is a large number of terms, as in this case, not all terms are written out explicitly; the three dots... symbol is used. Example(4) Find the common difference for each of the following arithmetic sequences. a. 4, 9, 4, b. 7, 5, 3, c. 3, 5 4, 9 2, d. 2, 9, 6, The common difference, d, in an arithmetic sequence is defined by: d = a n a n. a. 4, 9, 4, d = 4 9 = 5 b. 7, 5, 3, d = 5 7 = 2 c. 3, 5 4, 9 2, d = = = 3 4 d. 2, 9, 6, d = 9 2 = 7 pg. 4

5 Example(5) Find a formula of the n th term of the arithmetic sequence: 5,, 3, The n th term of any arithmetic sequence is defined by: a n = a + (n )d In this example: a = 5 and d = ( ) ( 5) = 4. Therefore, a n = 5 + (n ) 4 = 5 + 4(n ) Example(6) Find the sum of the first 0 terms of the arithmetic sequence: 3/2, 0, 3/2, 3,. The sum of the first n terms of any arithmetic sequence is given by S n = n 2 [2a + (n )d] In this example: a = 3 2 and d = (0) ( 3/2) = 3/2. Therefore, S 0 = 0 2 [2( 3/2) + (0 )(3/2)] = 05 2 pg. 5

6 Exercises (6.). Write the first four terms for the sequences specified by: a. a n = 4n b. a n = 9 2n c. a n = 5 n 2 2. Write each series in expanded form. a. 6 k k k! b. 8 k 3 c. 4 3 k+ d. 4 ( ) k x k k! 3. For each of the following arithmetic sequences, find the common difference, then write the next five terms: a. 9, 7/2, b. 2, 5, c. ln, ln 2, d. π, 3π, e. 0, 3, 6, f., 5/2, 4, 4. Given an arithmetic sequence with first term a = 8 and common difference d = 3, find the following terms: a. a 7 d. a Find the sum of the first 20 terms of the arithmetic series In an arithmetic series, the first term is 8, the last term is 65 and the sum is 730. Find the first 3 terms of the corresponding sequence of this series. 7. What is the n th term of the sequence: 4, 9/2, 5, pg. 6

7 6.2 Geometric Sequences and Series Summary GEOMETRIC SEQUENCES: A sequence {a n } is called Geometric Sequence if the quotient of any two successive terms a n and a n is a constant, r called Common Ratio and is given by: r = If the first term of a geometric sequence a and the common ratio d are known, then the n th term is defined by: a n = a r (n ) a n a n The sum of the first n terms of a geometric sequence is given by: S n = Σ n a r k = a r n r ; r Example() What is the common ratio of the geometric sequence, 3, 9, 27, 8, 243,. r = a n = 3 a n = 3 Example(2) What is the n th term of the geometric sequence: /2, /4, /8? In this geometric sequence: a = /2 and r = /4 /2 = /2. Therefore, the nth term is: a n = a r n = (/2)(/2) n = 2 n pg. 7

8 Example(3) What are the next three terms in the geometric sequence: 6, 4,,? In this sequence a = 6 and r = 4/6 = /4. Therefore, the n th term is given by: a n = 6( /4) n = ( /4) n 3 Now, to find the next 3 terms in the given series: a 4 = ( /4) 4 3 = /4 a 5 = ( /4) 5 3 = /6 Example(4) a 6 = ( /4) 6 3 = /64 Find the sum of the first 6 terms of the geometric sequence: 0/3, 50/9,. In this sequence: a = 0/3 and r = 5/3, therefore S 6 = a r n r = ( 0 3 ) ( ) (5/3) 6 (5/3) = Exercises (6.2). Find the n th term of each of the following geometric sequences: (a) /2,, 2, 4 (b), /2, /4, /8 (c) 4, 2, 36, Find the sum of the first 2 terms of the geometric series 4, 2, 36, 3. Find the sum of the series In a geometric sequence the fourth term is Find the ninth term of the sequence, 2, 2, 32 and the seventh term is. Find the common ratio of this sequence. 8 pg. 8