Sequences

Question 1

True or false?

Two different formulas can give sequences with equivalent terms

Answer 1

True

Question 2

Question 3

-1 to the power of what numbers equal 1?

(Even or odd)?

Answer 3

Even numbers

Think about squared and cubed, square a negative gives positive but cubed gives a negative

Question 4

-1 to the power of what numbers equal -1?

(Even or odd)?

Answer 4

Odd numbers

Think about squared and cubed, square a negative gives positive but cubed gives a negative

Question 5

Write the following in terms of a sequence

Answer 5

Basically it means, the sum of all a_k terms from where k=m (a_m) up until a_n

This is also called the expanded form

Question 6

In the summation notation below, what is n, k and m known as

Answer 6

• k = index of the summation
• m = lower limit of the summation
• n = upper limit of the summation

Question 7

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Question 8

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Question 9

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Question 10

Answer 10

We know that it ends at n because the last term of the sequence includes n

Question 11

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Question 12

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Question 13

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Question 14

Write the following in expanded form

Answer 14

Basically it means, the product of all a_k terms from where k=m (a_m) up until a_n

Question 15

Compute the following:

Answer 15

Question 16

Compute the following

Answer 16

Question 17

Given a_k and b_k are sequences of real numbers and n, k, m are integers.

Combine the following into one summation

Answer 17

Question 18

Given a_k are sequences of real numbers and n, k, m are integers and c is a real number

Simplify the following

Answer 18

Question 19

Given a_k and b_k are sequences of real numbers and n, k, m are integers.

Combine the following into one summation

Answer 19

Question 20

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Question 21

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Question 22

In the following, what is another name for k

Answer 22

It can also be called the dummy variable. It can be substituted as any other variable meaning:

Question 23

Is the following true?

Answer 23

Yes. This is because k is a dummy variable, it can be subsituted as any other variable. The left side is just k subsituted as (k-1)

Question 24

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Question 25

What is n!

Answer 25

n factorial. The product of all integers from 1 to n

Question 26

What is the difference between general/reccursive and explicit formula?

Answer 26

Explicit formulas directly calculate any term in a sequence given its position (n), while general formulas define a sequence recursively, meaning each term is defined based on the previous term(s).

Question 27

What is 0!

Answer 27

1

Question 28

Simplify then solve the following expression

Answer 28

Question 29

Simplify then solve the following expression

Answer 29

Question 30

Simplify then solve the following expression

Answer 30

Question 31

Simplify the following expression

Answer 31

Question 32

Simplify the following expression

Answer 32

Question 33

# True or false? A recurrence sequence not only depends on the equation but also the initial terms

Answer 33

True

Question 34

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Question 35

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Question 36

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Question 37

What are the three components a recursive definition consists of? Define them

Answer 37

I. Base: A statement that certain objects belong to a set II. Recursion: A collection of rules indicating how to form new set objects from those already known to be in the set III. Restriction: A statement that no objects occur in the set other than those formed from the recursion and defined by the base.

Question 38

Answer 38