Worksheet 66 (12.1) Chapter 12 Sequences and Series
12.1 Arithmetic Sequences
An infinite sequence is a function whose domain is the set of positive integers. It is expressed as a1, a2, a3, a4, ... , an, ...
Sequential functions are commonly represented by the variable a. The general term of the sequence is designated by an where an defines the rule or pattern for the sequence. The subscript n tells which term it is of the sequence. For example, a5 is the fifth term of the sequence. Note: Sequences eliminate the need for ordered pairs where the first coordinate tells which term, n, it is of the sequence and the second coordinate is the functional value of n, a(n) or an.
Finding the terms in a sequence:
1. Evaluate an when n = 1, n = 2, n = 3, etc. 2. List the functional values in order and separate them with a comma.
Summary 1:
Warm-up 1. a) Write the first five terms of the sequence that has the general term
24 an = 3n - 2. an = 3n - 2
Let n = 1: a1 = 3( ) - 2 = ______
Let n = 2: a2 = 3( ) - 2 = ______
Let n = 3: a3 = 3( ) - 2 = ______
Let n = 4: a4 = 3( ) - 2 = ______
Let n = 5: a5 = 3( ) - 2 = ______
The sequence is ____, ____, ____, ____, ____. Worksheet 66 (12.1)
b) Write the first three terms of the sequence that has the general term 2 an = 2n + 5. 2 an = 2n + 5
2 Let n = 1: a1 = 2( ) + 5 = ______
2 Let n = 2: a2 = 2( ) + 5 = ______
2 Let n = 3: a3 = 2( ) + 5 = ______
The sequence is ____, ____, ____.
c) Find the 150th term of the sequence an = -2n + 5.
Let n = 150: a150 = -2( ) + 5 = ______
n - 7 d) Find the fifth term of the sequence an = 3 .
____ - 7 Let n = 5: a5 = 3
= ______
Problems
25 1. Write the first five terms of the sequence that has the general term an = 5n -1.
2. Write the first three terms of the sequence that has the general term 2 an = -2n + 3.
3. Find the 35th term of the sequence an = 5n - 1.
Worksheet 66 (12.1)
n + 1 4. Find the 5th term of the sequence an = 2 .
An arithmetic sequence or arithmetic progression is a sequence where there is a common difference between successive terms. The arithmetic sequence exhibits constant growth.
The general term of an arithmetic sequence is given by the formula an = a1 + (n - 1)d where a1 is the first term in the sequence and d is the common difference.
Finding the general term of a given arithmetic sequence:
1. Find the common difference, d, in the sequence by subtracting two successive terms: a k + 1 - a k = d for every positive integer k. 2. Identify a1, the first term of the sequence. 3. Substitute a1 and d in the formula to define the general term of the given arithmetic sequence: a n = a 1 + (n - 1)d 4. This general term can now be used to find other indicated terms in the sequence.
Summary 2:
26 Warm-up 2. a) Find the general term of the arithmetic sequence 3, 0, -3, -6, -9, ...
Note: This is an example of an infinite arithmetic sequence.
Find the common difference, d: a k + 1 - a k = d 0 - 3 = d _____ = d
Identify a 1: _____ = a1
Substitute to define the general term: an = a1 + (n - 1)d
an = ( ) + (n - 1)( )
an = ______
Worksheet 66 (12.1)
b) Find the 21st term of the arithmetic sequence 3, 0, -3, -6, -9, ...
an = -3n + 6 (See above.) a21 = -3( ) + 6
= ______The 21st term is ______.
27 c) Find the general term of the arithmetic sequence -7, -2, 3, 8, 13, ...
Find the common difference, d: a k + 1 - a k = d -2 - (-7) = d _____ = d
Identify a 1: _____ = a1
Substitute to define the general term: a n = a 1 + (n - 1)d
an = ( ) + (n - 1)( )
an = ______
d) Find the 13th term of the sequence -7, -2, 3, 8, 13, ...
an = 5n - 12 (See above.)
a13 = 5( ) - 12
= ______The 13th term is ______.
e) Find the number of terms in the finite sequence -3, -1, 1, ..., 95.
Note: In a finite arithmetic sequence, the last term is an.
Find the common difference, d: a k + 1 - a k = d -1 - (-3) = d _____ = d
Identify a 1 and a n: _____ = a1 and _____ = an
Worksheet 66 (12.1)
Substitute to solve for n: a n = a 1 + (n - 1)d _____ = _____ + (n - 1)( )
n = ______
28 There are ______terms in the given finite sequence.
Problems
5. Find the general term of the arithmetic sequence 3, 1, -1, -3, -5, ...
6. Find the 40th term of the sequence 3, 1, -1, -3, -5, ...
7. Find the number of terms in the finite sequence -1, 4, 9, ..., 174.
Worksheet 67 (12.2)
12.2 Arithmetic Series
Summary 1:
29 A series is the indicated sum of a sequence. The sum of the first n terms of an arithmetic series is given by
n ( a1 + an ) S n = , where n tells the number of terms, a1 is the first term of the 2 sequence, and an is the last term of the sequence.
Finding the sum of a given arithmetic sequence:
1. Identify a1, n, and d for the sequence.
2. Find an using an = a1 + (n - 1)d.
n ( a1 + an ) 3. Substitute and evaluate: S n = 2
Warm-up 1. a) Find the sum of the first 60 terms of 2 + 4 + 6 + 8 + ...
Note: This is an example of an infinite series.
Identify a 1, n, and d for the sequence: a1 = ______, n = ______, d = ______
Find a n: an = a1 + (n - 1)d a60 = _____ + ( _____ - 1)( ) a60 = ______
Substitute and evaluate S n:
30 n ( a1 + an ) S n = 2 ( )( + ) S 60 = 2
S 60 = The sum of the first 60 terms is ______. Worksheet 67 (12.2)
b) Find the sum of all multiples of 5 between 13 and 601. The series is _____ + _____ + _____ + ... + _____.
Note: This is an example of a finite series.
Identify a 1, a n, and d to find n: a1 = ______, an = ______, d = ______
Find n: an = a1 + (n - 1)d _____ = _____ + (n - 1)( )
n = ______
Substitute and evaluate S n:
n ( a1 + an ) S n = 2 ( )( + ) S118 = 2
S118 = The sum of the multiples of 5 between 13 and 601 is ______.
c) Find the sum of the first 40 terms of the series with an = 2n + 3.
Identify a 1, n, d, and a n: a1 = 2( ) + 3; a2 = 2( ) + 3; a40 = 2( ) + 3 a1 = ______a2 = ______a40 = ______
The series is _____ + _____ + ... + _____. a1 = ______, n = ______, d = ______, an = ______
Substitute and evaluate S n:
n ( a1 + an ) S n = 2 ( )( + ) S 40 = 2
S 40 = The sum is ______. Worksheet 67
31 (12.2)
d) A beginning teacher's salary in 1978 was $8,500 per year. If this salary were to be consistently raised by $850 per year for 20 years, what would the total earnings be?
an = a1 + (n - 1)d a20 = ______+ ( _____ - 1)( )
a20 = ______
The series is ______+ ______+ ... + ______.
n ( a1 + an ) S n = 2 ( )( + ) S 20 = 2
S 20 = The total earnings would be ______.
Problems 1. Find the sum of the first 50 terms of 3, 6, 9, 12, ...
2. Find the sum of all multiples of 4 between 15 and 601.
Worksheet 68 (12.3)
12.3 Geometric Sequences and Series
32 A geometric sequence or geometric progression is a sequence in which each term after the first is obtained by multiplying the preceding term by a common multiplier.
The common ratio of a sequence is the common multiplier.
n - 1 The general term of a geometric sequence is given by an = a1 r where a1 is the first term and r is the common ratio.
Finding the general term of a given geometric sequence:
1. Find the common ratio, r, in the sequence by dividing two successive
a k + 1 terms: = r for every positive integer k. a k
2. Identify a1, the first term of the given sequence.
3. Substitute a1 and r in the formula to define the general term of the n - 1 given geometric sequence: a n = a 1 r 4. The general term can now be used to find other indicated terms in the sequence.
Summary 1:
33 Warm-up 1. a) Find the general term for the geometric sequence 3, 9, 27, 81, ...
Find the common ratio, r:
a k + 1 = r a k 9 = r 3 r =
Identify a 1: a1 = ______
Substitute to define the general term: n - 1 a n = a 1 r Worksheet 68 (12.3)
n - 1 a n = ( )( )
an = ______
b) Find the 10th term of the preceding sequence:
n an = 3 ( ) a10 = 3 a10 = ______
c) Find the general term for the geometric sequence: -50, 10, -2, 2/5, -2/25, ...
Find the common ratio, r:
a k + 1 = r a k 10 = r - 50 r =
Identify a 1: a1 = ______
Substitute to define the general term: n - 1 a n = a 1 r n - 1 a n = ( )( )
an = ______
34 d) Find the eighth term of the preceding sequence: n - 1 1 a n = (-50) - 5 ( ) - 1 1 a8 = ( - 50 ) - 5
a8 = Worksheet 68 (12.3)
Problems
1. Find the general term for the geometric sequence -1, -2, -4, -8, ...
2. Find the 12th term of the preceding sequence.
3. Find the general term for the geometric sequence -16, 4, -1, 1/4, ...
4. Find the 9th term of the preceding sequence.
Summary 2:
35 A geometric series is the indicated sum of a geometric sequence. The sum of the first n terms of a geometric series is given by n a 1 r - a 1 S n = where n tells the number of terms , a1 is the first term of the r - 1 sequence, and r 1.
Finding the sum of the first n terms of a geometric sequence:
1. Identify a1, r, and n for the sequence. n a 1 r - a 1 2. Substitute and evaluate: S n = r - 1
Warm-up 2. a) Find the sum of the first 10 terms of -5 + 10 + (-20) + 40 + ...
Identify a 1, r, and n: a1 = _____, r = _____, n = _____ Worksheet 68 (12.3)
Substitute and evaluate: n a 1 r - a 1 S n = r - 1 ( )( )( ) - ( ) S 10 = ( ) - 1
S 10 = b) Find the sum of the first 6 terms of 6 + 3 + 3/2 + 3/4 + ...
36 Identify a 1, r, and n: a1 = _____, r = _____, n = _____
Substitute and evaluate: n a 1 r - a 1 S n = r - 1 ( )( )( ) - ( ) S 6 = ( ) - 1
S 6 =
Problems
5. Find the sum of the first 9 terms of 5 + 10 + 20 + 40 + ...
6. Find the sum of the first 7 terms of -10, 5, -5/2, 5/4, ...
Worksheet 69 (12.4)
12.4 Infinite Geometric Series
Summary 1:
37 n In general,where a 1for is thevalues first of term r such and that rr < 1<. 1, the expression r will approach 0 as n increases. As a result, the formula for the sum of an infinite n For values of r such that r 1, the expression r a increases 1 without bound. As a geometric series can be expressed as S = result, the sum of such an infinite geometric series does1 - r not exist.
Warm-up 1. Find the sum of the infinite geometric series:
a) 1, 1/4, 1/16, 1/64, ...
Find r and determine the true statement about r: ( ) = r ( ) 1 4 = r Circle the true statement about r:
a 1 r < 1; therefore, S = . 1 - r r 1; therefore, the sum does not exist.
Substitute and evaluate:
a 1 S = 1 - r ( ) S = 1 - ( )
4 S = 3
Worksheet 69 (12.4)
b) 3, 9, 27, 81, ...
Find r and determine the true statement about r:
38 ( ) = r ( ) 3 = r Circle the true statement about r:
a 1 r < 1; therefore, S = . 1 - r r 1; therefore, the sum does not exist.
Since r 1, the sum ______.
Problems - Find the sum of the infinite geometric series.
1. 2, 2/3, 2/9, 2/27, ...
2. 1, 4, 16, 64, ...
Worksheet 70 (12.5)
12.5 Binomial Expansions
Summary 1:
39 n! = n (n - 1) (n - 2) (n - 3) 1, where Factorial n is any positive integer. Ex: 6! = 6 5 4 3 2 1 = 720 Note: 0! = 1 and 1! = 1
Warm-up 1. Find the value of 4!
4! = 4 ___ ___ 1 = ____
Summary 2:
Binomial Expansion
n n(n -1) 2 n(n -1)(n - 2) 3 (x+ y ) = xn + nxn-1 y + xn-2 y + xn-3 y + + ysupn 2! 3!
Warm-up 2. Expand (2a - b)5 ` Let x = _____ and y = _____
= ( )a5 - ( )a4 b + 80( )b2 - ( ) + ( )ab4 - ( )
40 = ______
Worksheet 70 (12.5)
Problems:
1. Expand and simplify: (2x + y)5
2. Expand and simplify: (a - 2)6
Finding Specific Terms
A specific term of a binomial expansion may be found without writing the entire expansion. To find a specific term of (x + y)n: 1. Let r = one less than the number of the specific term. 2. The exponent of y for the specific term is r. 3. The exponent of x for the specific term is (n-r). 4. The numerator of the coefficient contains r factors where the first is n and each succeeding factor is one less than the preceding one. 5. The denominator is r! 6. Put the information together and simplify.
Summary 3:
41 Warm-up 3. Find the 7th term of (2a - b)10
Let x = 2a, y = -b, and n = 10
1) r = 7 - 1 = ___ 2) The exponent of y is r = ___ 3) The exponent of x is (n-r) = (10 - ___) = (___) 4) The numerator of the coefficient is ___9_________5 5) The denominator of the coefficient is r! = 6! ( ) 9 ( )( )( ) 5 6) (2a )( )( )6 6! = ( )16a4b6 = ( )a4b6 Worksheet 70 (12.5)
Problems: 3. Find the 3rd term of (x + 5)6
4. Find the 4th term of (3x - 4)5
42