Chapter 7 - Sequences

Question 1

Definition of Sequence (4)

Answer 1

• Ordered list of numbers that follow a specific rule/pattern
• Can be represented by a formula, infinite or finite
• Domain (Position) & Range (Term) goes on forever
• Formally, it’s a list of ordered pairs: (1, a₁), (2, a₂), (3, a₃), …

Question 2

Definition of Arithmetic Sequence

Answer 2

List of number that get added by the same number (d)

Question 3

What is d? How do you find it? (2)

Answer 3

1. Common Difference (What is continuously added to terms in an arithmetic sequence)
2. To Find:
   - Use Slope (Term₁- Term₂)/ (Position₁- Position₂)
   - Plug into arithmetic equation

Question 4

What is the formula for an Arithmetic Sequence? (3)

Answer 4

1. aₙ = a₁ + (n - 1) (d)
2. Representations:
   aₙ = Term in Position (n)
   n = Position number of Term
   d = common difference
3. Use if you need to find first term, common difference, position of term, or term when given a position

Question 5

Definition of Geometric Sequence

Answer 5

List of numbers that get multiplied by same number (r)

Question 6

What is r? How do you find it? (1/3)

Answer 6

1. Common ratio between successive terms
   (Term 2/Term 1 = Term 3/Term 2)
2. To Find:
   - Plug into geometric equation
   - Place Quotient (Term/Term) under root (distance between positions)
   - If root is even, add positive negative to r

Question 7

What is the formula for Geometric Sequence? (3)

Answer 7

1. aₙ = a₁ · r ⁿ⁻¹
2. Representations:
   aₙ = Term in Position (n)
   n = Position of Term
   r = common ratio
3. Use if you need to find first term, position of term or term when given position

Question 8

Definition of Arithmetic Means, how do you find it? (2)
Ex: Arithmetic Means between two numbers

Answer 8

1. Number in sequence is avg of 2 numbers on its L/R
2. Plug two numbers as slope to solve for d (continue on to find numbers between the two numbers)

Question 9

Definition of Geometric Means, how do you find it? (3)

Answer 9

Def: Number in sequence is square root of product of factors of 2 numbers on its L/R
Find:
- Root (distance of numbers) of Quotient to solve for r
- Multiply r to each succesive term

Question 10

Definition of Arithmetic Sum Formula (2)

Answer 10

1. Sₙ = n (a₁ + aₙ)/2
   Sₙ = sum
   n = number of terms in sum
   a₁ = starting term
   aₙ = last term of sum (use arithmetic formula to solve)
2. If missing any information, use arithmetic explicit formula to find vairable to solve for sum

Question 11

Definition of Geometric Sum Formula (2)

Answer 11

1. Sₙ =a₁ (1 - rⁿ)/(1 - r)
   Sₙ = sum
   n = number of terms in sum
   a₁ = starting term
   r = Common Ratio (Root of Quotient)
2. If missing any information, use geometric explicit formula to find variable to solve for sum

Question 12

Definition of Explicit Form (3)

Answer 12

1. Formula that directly gives value of nth term w/o needing to know previous terms
2. aₙ = a₁ + (n - 1) (d)
3. aₙ = a₁ · r ⁿ⁻¹

Question 13

Definition of Geometric/Arithmetic Series

Answer 13

Arithmetic Series: Sum of terms in an arithmetic sequence
Geometric Series: Sum of terms in a geometric sequence

Question 14

What is this notation: (Number) !

Answer 14

Factorial: Multiply number by every whole number less than it, down to 1

Question 15

Definition of Summation (2)

Answer 15

• Process of adding a sequence of numbers together
• Represented using Sigma (Σ) notation

Question 16

What is summation notation?

Answer 16

• Short way of writing sum of many terms using Σ
• Sum is essentially includes terms at and between n and z
• Arithmetic/Geometric Explicit Formula is in parentheses

(x) ᶻΣ (_____)
n = ___

z = ending position of sum
n = starting position of sum
(____) = Geometric/Arithmetic Explicit Formula

Question 17

When given a series of numbers, how do you rewrite as summation?

Answer 17

1. Immediately assume starting position is 1 (n = 1)
2. Ending position is just the position of final term
3. Find Arithmetic/Geometric Explicit Formula
4. Plug in a₁ & d/r, and simplify
5. If it’s not either (no common difference/common ratio), try to find a pattern to apply to each term

ᵉⁿᵈᶦⁿᵍ ᵖᵒˢᶦᵗᶦᵒⁿ Σ (Explicit Formula)
n = 1

Question 18

How do you find the sum of numbers when given summation notation (3)

Answer 18

1. Identify starting and ending positions
2. Plug each position into the explicit formula
3. Add up all results of explicit formula in a sum

Question 19

What is recursive definition?

Answer 19

1. Describes how each new term is built from earlier terms
2. Uses a starting point equation (defines a₁)
3. Ask yourself: What changes are being applied to previous term?

Question 20

What is the notation for recursive definition? (3)

Answer 20

1. ALWAYS: aₙ = aₙ ₋₁ _______, a₁ = _
2. Base Case: Provides starting point, defines a₁ (a₁ = _)
3. Recursive step: aₙ = aₙ ₋₁ _______, blank is what change to previous term is applied to achieve the next term (Arithmetic: +3, Geometric = ·2, Pattern:

Question 21

What is something to keep in mind in notation for recursive definition? (2)

Answer 21

1. If more than one previous term is needed to show how new term was built
2. Must provide more than one starting points (a₁ = _, a₂ = _)

Question 22

If there is one thing to know about recursive definitions, what should you know?

Answer 22

ALWAYS: aₙ = aₙ ₋₁ _______, a₁ = _
1. The variable part of the notation is the change that was applied to the previous term in order to achieve the next term
2. For example, if the sum of last two terms produced the next term, it would be: aₙ = aₙ ₋₁ + aₙ ₋₂
3.

Question 23

How do you translate a series into a recursive definition? (3)

Answer 23

1. Always start with this notation:
   aₙ = aₙ ₋₁ _______
   a₁ = _
2. Figure out if geometric, arithmetic, or pattern
3. Apply that change by adding or multiplying it to the previous term
4. Define a1, and determine if you need another term to solve for next term

Question 24

What is the difference between explicit formula and recursive definition?

Answer 24

Explicit Formula: Direct formula that shows you how to find the nth term
Recursive Definition: Describes how each new term builds off the previous term

Question 25

Using this formula, answer the following questions: Aₚ = R · [(1 - (1 + i)⁻ⁿ)/i] What does each variable stand for?

Answer 25

Aₚ = Loan Size R = Recurring Payment (fixed payment you must pay per unit of time) i = monthly interest rate (interest rate/each payment period) n = total number of payments (number of years x number of months)

Question 26

Using this formula, answer the following questions: Aₚ = R · [(1 - (1 + i)⁻ⁿ)/i] What is a down payment and how do you apply it? (2)

Answer 26

1. Initial amount of money you pay upfront when purchasing something expensive before financing rest through a loan 2. Subtract downpayment from loan to plug in Aₚ

Question 27

Using this formula, answer the following questions: Aₚ = R · [(1 - (1 + i)⁻ⁿ)/i] How do you find the total amount paid?

Answer 27

Multiply the recurring payment (R) by the number of payments (n)

Question 28

Using this formula, answer the following questions: Aₚ = R · [(1 - (1 + i)⁻ⁿ)/i] How do you find how much is paid in interest? (2)

Answer 28

1. Difference between total amount paid and selling price 2. Total amount paid: R (n)

Question 29

What is an infinite sum?

Answer 29

Sum of infinite number of terms

Question 30

In what cases will there be no sum if the sum has an infinite number? What cases will there be a sum? (2)

Answer 30

1. In arithmetic and exponential functions, if they do not approach an asymptote, they will just approach infinity (or negative infinity) and there will be no sum 2. For a decreasing exponential function, the values will approach an asymptote if "r" in the geometric series is between -1 and 1

Question 31

What do they mean if they ask you to find the limit a function approaches? (2)

Answer 31

1. Value partial sums approach as number of terms goes to infinity 2. If r is between -1 and 1, (decreasing exponential/geometric series), limit is given by infinite sum formula

Question 32

What is the formula for infinite sum? (2)

Answer 32

1. Only use if geometric and -1 < r < 1 2. S∞ = a₁/1 - r

Question 33

How do you rewrite a decimal as a fraction? (4)

Answer 33

1. Use infinite sum formula (S∞ = a₁/1 - r) 2. Identify repeating part of decimal = a1 3. Find common ratio (1/10ᵏ), k = number of repeating digits 4. Plug into formula & simplify fraction

Question 34

How do you check your sum?

Answer 34

Alpha Window 2: Plug in stuff into sigma notation to find sum

Question 35

How do you find a pattern if it is not arithmetic or geometric? (7)

Answer 35

Explicit Formula 1. Write positions under terms 2. Find what changes to apply to n (position) to turn your position into term (Note: If fraction, treat numerator then treat denominator) Recursive Definition 1. Use previous term (treat as a_n-1) 2. Find what changes to apply to turn previous term into next term -------------------------------------------------------------------------------------- - Helps to identify if a function is linear/exponential/quadratic - Plug into equations to test candidate formula - Use squares, cubes, factorials, multiplying, dividing, adding and subtracting

Question 36

When asking about money increase (in salary), What do you do if they ask something increases by a percentage every year? (3)

Answer 36

- Convert percentage into decimal - ADD ONE to make sure it is increasing - Substitute in for r (typically use sum formula)

Question 37

What do you do if there is an even root when solving for r? (2)

Answer 37

- Add positive negative to "r" (ONLY in geometric sequence - There will be two different sequences

Question 38

Word Problem relating to Salary: (3/2) 1. What will salary be in given years? 2. What will be total amount in given years?

Answer 38

1. Salary in "__" years: Solve for aₙ - Use geometric explicit formula (aₙ = a1 (r)ⁿ⁻¹) - Plug rate in for "r" (increases by _%, divide by 100 & add 1 to make money increase) - Starting Salary is a₁, plug in rate for "r", plug in number of years for "n" 2. Total amount in "__" years: Solve for Sn - Use geometric sum formula (a₁ (1 - rⁿ)/(1 - r)) - Plug in years into n, rate into r (divide by 100, add 1), starting salary for a1