Hon Alg 2: Unit 6



Hon Math III: Unit 6

SEQUENCES: A sequence is an ordered list of numbers.

GENERAL NOTATION: The numbers of a sequence are called TERMS

• “a” refers to the NUMERICAL VALUE of a term in the sequence

• Subscript of “a” refers to the LOCATION of a term in the sequence.

Terms of a sequence generally follow a pattern or operations to move forward to the next term.

a1 |a2 |a3 |a4 |a5 |a6 |… |an | |

To move backward between terms, we use OPPOSITE operations

EXAMPLES: Determine if each sequence is Arithmetic or Geometric

i. Write the recursive rule for each sequence (an = …)

ii. Find the NEXT TWO TERMS of the sequence based on your pattern

(1) 5, 8, 11, 14, 17, …

(2) 3, 6, 12, 24, …

(3) 19, 12, 5, -2, -9, …

(4) -2, 3, - 4.5, 6.75, …

(5) 8.3, 10.9, 13.5, 16.1,, …

(6) -4, -20, -100, -500, …

(7) 3.5, 7.35, 15.435, 32.4135, …

(8) 237, 203, 169, 135, 101, …

(9) 7, -8.4, 10.08, -12.096, …

(10) [pic], 1, [pic], [pic], [pic], [pic], …

(11) [pic], [pic], [pic], [pic] …

(12) 2x, 6x3, 18x5, 54x7, …

(13) 23x, 36x, 49x, 62x, 75x, …

ARITHMETIC SEQUENCES:

The difference between consecutive terms is constant value.

• Patterns of ADD or SUBTRACT by a constant value.

• The common difference, d, is constant difference.

o d is POSITIVE when the pattern is _________

o d is NEGATIVE when the pattern is _________

How do you find the common difference?

GEOMETRIC SEQUENCES:

The consecutive terms differ by a constant factor.

• Patterns of MULTIPICATION by a constant value.

o Reminder: Division ( Multiplication by Reciprocal

• The COMMON RATIO, r, is constant factor.

How do you find the common ratio?

How might you find a15 in each of these sequences? (Hint: Are they geometric or arithmetic?)

1) an = an – 1 + 6, a1 = 7

2) an = an – 1 - 3.5, a1 = 47

3) 108, 117, 126, 135, …

4) an = 1.5an – 1, a1 = 8

5) an = 0.75an – 1 , a1 = 576

6) [pic]

EXPLICIT FORMULAS: Directly finds any term of a sequence given the term number (location) in the sequence

nth term of an arithmetic sequence: an = a1 + d(n – 1)

d = common difference

a1 = first term n – 1 = how many terms a1 and an are apart

(7) a1 = 2, d = 7, n = 14

(8) a1 = 35, d = 2.3, n = 8

(9) 121, 118, 115, … a21 = ?

nth term of a geometric sequence: an = a1 (r)n – 1

r = common ratio

a1 = first term n – 1 = how many terms a1 and an are apart

(10) a1 = 16807, r =[pic], n = 6

(11) 160, 240, 360, … a9 = ?

(12) a1 = 2, r = 5, n = 7

Hon Math III: Unit 6 Name: ____________________________

ARITHMETIC SEQUENCES:

PRACTICE: Find the indicated nth term and write an equation for the nth term of the sequence.

(1) -17, -13, -9, … a12 = ?

(2) a1 = 17, d = -9, n = 20

(3) 3.46, 3.73, 4, … a17 =?

Example: Find the missing terms of an arithmetic sequence (arithmetic means)

(4) Find the 5 arithmetic means: a1 = 3, _____, ______, ______, ______, ______, a7 = 27

What is the total change in the values of the given terms?

How far apart are the locations of the given terms?

What is the common difference (individual change between terms)?

(5) Find 3 arithmetic means of 24 and 76 (6) Find 4 arithmetic means of 57 and 38

Arithmetic Sequences with non-consecutive terms given?

Calculate the value of the common difference, Find the a1 value, Find the an.

(7) a1 = 16 and a6 = 91, a11 = ?

(8) a1 = 10 and a5 = -14, a3 = ?

(9) a3 = - 8 and a8 = 7, a5 = ?

(10) a5 = 9107 and a18 = 11928, a10 = ?

Finding the n for a given value in an arithmetic sequence.

(11) Find n, if an = 633, a1 = 9, and d = 24.

(12) 4, 7, 10, 13, … find n for term = 301

(13) 89, 72, 55, 38, … find n for an = - 115

(14) Find n, if an = 30.06, a1 = 14.38, d = 0.56.

Hon Math III: Unit 6 Name: ____________________________

SUMMATION or SERIES (Sn): The sum of consecutive terms from a sequence.

Sn = a1 + a2 + a3 + …+ an-1 + an Summation (SIGMA) Notation: [pic]

Exp #1. [pic] Exp #2. [pic]

What terms from the given sequence should be added based on the summation notation below?

(1) [pic] (2) [pic] (3) [pic]

Identify the type of sequence (arithmetic or geometric) and what terms should be added?

(4) [pic] (5) [pic] (6) [pic]

(7) [pic] (8) [pic] (9) [pic]

ARITHMETIC SERIES (SUM, Sn):

The sum of consecutive terms for an arithmetic sequence.

EXAMPLE: Find the sum of the arithmetic sequence 4, 11, 18, 25, 32, 39, 46, 53, 60.

Formula for ARITHMETIC SERIES (SUM): [pic]

Alternative: [pic]

(10) Find the arithmetic series of …

a. the first 100 natural numbers. b. -3.4 – 1.6 + 0.2 + 2 + 3.8 + 5.6 + 7.4 + 9.2

c. 23 + 32 + 41 + 50 + 59 + 68 + 77 + 86 + 95 + 104 + 113

(11) Find Sn for the described arithmetic series

a. a1 = 25, an = 259, n = 14

b. a1 = 188, an = 34, n = 23

c. a1 = 14.72, an = 24.41, n = 18

(12) Find the sum of the arithmetic sequence

a. S50 of 29, 44, 59, 74, 89, …

b. S63 of -19, -13, -7, - 1, …

c. S42 of 324, 351, 378, 405, …

(13) Find the sum of the arithmetic series

a. 6 + 27 + 48 + … + 363

b. 17 + 23 + 29 + … + 131

c. 8012 + 7999 + 7986 +…+ 6075

(14) Find n for the given descriptions

a. a1 = 5, an = 77, Sn = 1025

b. a1 = 49, an = -103, Sn = -540

c. a1 = 13.7, an = 164.1, Sn = 1511.3

d. a1 = 5, d = 3, Sn = 440

e. a1 = 4, d = 9, Sn = 58

f. a1 = 234, d = - 18, Sn = 1620

Hon Math III: Unit 6 Name: ____________________________

GEOMETRIC SEQUENCES

PRACTICE: Find the indicated nth term and write an equation for the nth term of the sequence.

(1) a1 = [pic], r = 3, n = 8

(2) [pic], [pic], [pic], … a6 = ?

(3) 4, -12, 36, … a8 = ?

Example: Find the missing terms of a geometric sequence (geometric means)

(4) Find 3 geometric means of a1 = 2.25, _________, _________, __________, a5 = 576

i. How many multiplications between the terms?

ii. What is a possible common ratio?

(5) Find 2 geometric means of 24 and 81. (6) Find 3 geometric means of 324 and 64

(7) Find 3 geometric means of a1 = 162 and a5 = 32 (8) Find 2 geometric means of a1 = -2 and a4 = 54

Geometric Sequences with non-consecutive terms given?

(Step1) Calculate or identify the value of the common ratio (Step 2) Find the a1. (Step 3) Find the an.

(9) a4 = 27, r = [pic], find a7

(10) a3 = 180, r = - 6, find a6

(11) [pic] find a2

(12) a2 = 8, a3 = 2, find a5

(13) a4 = 192, a6 = 3072, find a9

(14)[pic]find a1

Hon Math III: Unit 6 Name: ____________________________

GEOMETRIC SERIES (SUM, Sn): Sum of consecutive terms for a geometric sequence.

FORMULA: a1 + a2 + a3 + a4 +…+ an-1 + an = [pic]

PROOF of FORMULA:

Sn = a1 + a2 + a3 + a4 +…+ an-1 + an every term of the geometric sequence is an = a1 (r)n – 1

Sn = a1 + a1r + a1r2 + a1r3 +…+ a1rn-2 + a1rn-1 Multiply sum by common ratio, r

r • (Sn = a1 + a1(r) + a1(r) 2 + a1(r) 3 +…+ a1(r) n-2 + a1(r) n-1) Distribute r

rSn = a1(r) + a1(r) 2 + a1(r) 3 + a1(r)4 + …+ a1(r) n-1 + a1(r) n Subtract Sn and rSn

Sn – rSn = a1 – a1(r)n Combine “cancel” like terms

Sn (1 – r) = a1 ( 1 – rn) Factor GCF

[pic] Alternative Formula: [pic]

(1) Find the geometric sum of …

a. 3 + 6 + 12 + 24 + 48 + 96 + 192 + 384 =

c. 7 + 21 + 63 + … to 10 terms =

b. [pic] + [pic] + 1 + … to 7 terms =

d. [pic] - [pic] + 20 - … to 6 terms =

e. a1 = 162, r = 1/3;

S6 = ???

f. a1 = 3, a8 = 384, r = 2

S8 = ???

g. a1 = 5, r = 3

S12 = ?

(2) Find the indicated geometric series Sn

a. a1 = 625, r = 3/5;

S5 = ?

b. a2 = -36, a5 = 972;

S7 = ?

c. a1 = 1296, an = 1, r = -1/6

Sn = ???

(3) Find the first term for each geometric series

a. S8 = 39,360, r = 3, a1 = ??

b. S6 = -364, r = -3, a1 = ??

c. S7 = [pic], r = [pic], a1 = ??

d. S6 = [pic], r = [pic], a1 = ??

e. r = 2, Sn = 189, an = 96 f. r = 3/2, Sn = 43134.75, an = 14762.25

a1 = ?? a1 = ??

INFINITE GEOMETRIC SERIES (SUM, Sn): The sum, S, of an infinite geometric series in which the common ratio is between -1 and 1 ( - 1 ≤ r ≤ 1)

S = a1 + a2 + a3 + a4 + … = [pic]

PRACTICE: Determine if an infinite geometric series exists or not, if yes find the sum.

• Hint: Find the common ratio

(1) [pic]

(2) [pic]

(3) [pic]

(4) [pic]

(5) [pic]

(6) [pic]

(7) - 3 – 4.5 – 6.75 – …

(8) -3 – 1.8 – 1.08 - …

(9) – 64 + 48 – 36 + 27 - …

REPEATING DECIMALS: Write each repeating decimal as a fraction

Example: [pic] = 0.25 + 0.0025 + 0.000025 + 0.00000025 + …

Infinite geometric series: [pic], [pic], so [pic]

1) [pic]

2) [pic]

3) [pic]

4) [pic]

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