Worksheet 66 (12.1)

Chapter 12 Sequences and Series

12.1 Arithmetic Sequences

Summary 1:

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.

280

Warm-up 1. a) Write the first five terms of the sequence that has the general term 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 an = 2n2 + 5.

an = 2n2 + 5

Let n = 1: a1 = 2( )2 + 5 = _______

Let n = 2: a2 = 2( )2 + 5 = _______

Let n = 3: a3 = 2( )2 + 5 = _______

The sequence is ____, ____, ____.

c) Find the 150th term of the sequence an = -2n + 5.

Let n = 150: a150 = -2( ) + 5 = _______

d) Find the fifth term of the sequence an = 3 n - 7.

281

Let n = 5: a5 = 3 ____ - 7

= _______

Problems

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

an = -2n2 + 3.

3. Find the 35th term of the sequence an = 5n - 1.

Worksheet 66 (12.1)

4. Find the 5th term of the sequence an = 2 n + 1.

Summary 2:

282

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

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.

283

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 _______.

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, -

284

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 = _______

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.

285

Worksheet 67 (12.2)

12.2 Arithmetic Series

Summary 1:

A series is the indicated sum of a sequence. The sum of the first n terms of an arithmetic series is given

by , where n tells the number of terms, a1 is the first term of the 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.3. Substitute and evaluate:

286

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:

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:

287

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:

The sum is _______.

Worksheet 67 (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 __________ + __________ + ... + __________.

The total earnings would be

288

____________.

Problems1. 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

Summary 1:

289

Warm-up 1. a) Find the general term for the geometric sequence 3, 9, 27, 81, ...

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.

The general term of a geometric sequence is given by an = a1 r n - 1

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 terms: for every positive integer 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 given geometric sequence:

4. The general term can now be used to find other indicated terms in the sequence.

290

Find the common ratio, r:

Identify a 1: a1 = _______

Substitute to define the general term:

Worksheet 68 (12.3)

an = ________

b) Find the 10th term of the preceding sequence:

an = 3n

a10 = 3( )

a10 = _________

c) Find the general term for the geometric sequence: -50, 10, -2, 2/5, -2/25, ...

Find the common ratio, r:

Identify a 1: a1 = _______

Substitute to define the general term:

an = ___________

d) Find the eighth term of the preceding sequence:

291

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:

292

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:

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

where n tells the number of terms , a1 is the first term of the 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.2. Substitute and evaluate:

293

b) Find the sum of the first 6 terms of 6 + 3 + 3/2 + 3/4 + ...

Identify a 1, r, and n:a1 = _____, r = _____, n = _____

Substitute and evaluate:

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:

294

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:

Circle the true statement about r:

r < 1; therefore, .r 1; therefore, the sum does not exist.

Substitute and evaluate:

Worksheet 69 (12.4)

b) 3, 9, 27, 81, ...

Find r and determine the true statement about r:

In general, for values of r such that r < 1, the expression rn will approach 0 as n increases. As a result, the formula for the sum of an infinite geometric series can be expressed as

where a1 is the first term and r < 1.

For values of r such that r 1, the expression rn increases without bound. As a result, the sum of such an infinite geometric series does not exist.

295

Circle the true statement about r:

r < 1; therefore, .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 ExpansionsSummary 1:

296

Warm-up 1. Find the value of 4!

4! = 4 ___ ___ 1 = ____

Summary 2:

Warm-up 2. Expand (2a - b)5

`Let x = _____ and y = _____

Binomial Expansion

n! = n (n - 1) (n - 2) (n - 3) 1, where n is any positive integer.Ex: 6! = 6 5 4 3 2 1 = 720

Note: 0! = 1 and 1! = 1

Factorial

297

= ____________

Worksheet 70 (12.5)

Problems:

1. Expand and simplify: (2x + y)5

2. Expand and simplify: (a - 2)6

Summary 3:

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.

Finding Specific Terms

298

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!

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

299