Mathematical Patterns

Question 1

Fibonacci Sequence

Answer 1

0, 1, 1, 2, 3, 5, 8, 13, 21, 34

Question 2

triangle sequences

Answer 2

1, 3, 6, 10, 15

Question 3

geometric sequences

Answer 3

2, 4, 8, 16, 32

Question 4

arithmetic sequence

Answer 4

2, 4, 6, 8

Question 5

How do you know if a number is consecutive?

Answer 5

A number is consecutive if it is right next to, either before or after, another term. For example, the consecutive number after 10 is 11. The 11 is right next to the 10. The consecutive number before 10 is 9.

Question 6

What are the first 10 consecutive numbers?

Answer 6

Because no special number set is mentioned, the first 10 consecutive numbers refer to the first ten terms beginning with 1.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Question 7

What is consecutive number example?

Answer 7

A consecutive number example is 11 and 12. Another question might be, what is the next consecutive even number after 9. That number is 10. 9 is not an even number, but the next closes even number after 9 is 10. That makes it the next consecutive even number after 9. The consecutive even number after 10 is 12.

Question 8

How do you find consecutive numbers?

Answer 8

The easiest way to find consecutive numbers is to look at the number set and find the number that is right next to the term of interest, either before or after.

Question 9

Consecutive Numbers

Answer 9

Consecutive numbers are numbers that go in order. For example, {5, 6, 7, 8, 9, 10} is a set of consecutive numbers. We can use a formula to determine the sum of any amount of consecutive numbers. Let’s practice some problems finding the sum of consecutive numbers.

Question 10

Determine the sum of the first 10 numbers of the set {5, 6, 7, 8, 9, 10, …}.

Answer 10

The first number is 5 while the 10th number is 14. Using n = 10, the sum is

(10/2)(5 + 14) = (5)(19) = 95

Question 11

Determine the sum of the numbers of the set {100 - 120}.

Answer 11

There are 21 numbers between 100 and 120 so n = 21. So, the sum is

(21/2)(100 + 120) = (10.5)(220) = 2,310

Question 12

Determine the sum of the numbers of the set {1000 - 1100}

Answer 12

There are 101 numbers between 1000 and 1100 so n = 101. So, the sum is

(101/2)(1000 + 1100) = (50.5)(2100) = 106,050

Question 13

Determine the sum of the first 1000 numbers of the set {2 - 1004}.

Answer 13

Here, the first number is 2 while the 1000th number is 1001. Using n = 1000, the sum is

(1000/2)(2 + 1001) = (500)(1003) = 501,500

Question 14

Determine the sum of the numbers of the set {1000 - 2000}.

Answer 14

There are 1001 numbers between 1000 and 2000 so n = 1001. So, the sum is

(1001/2)(1000 + 2000) = (500.5)(3000) = 1,501,500

Question 15

What are the patterns in Pascal’s triangle?

Answer 15

There are many patterns within Pascal’s triangle. The simplest pattern is that each number (apart from the outer 1’s) can be found by adding the two numbers above it. Natural numbers and triangular numbers appear along the diagonals, to name just two other patterns.

Question 16

What is Pascal’s triangle used for?

Answer 16

Pascal’s triangle can be used to quickly look up the values of the binomial coefficients. This appear in the expansion of a power of a binomial, and are otherwise tedious to calculate. The binomial coefficients often appear separately in combinatorics.

Question 17

What is Pascal’s triangle in mathematics?

Answer 17

Pascal’s triangle is an array containing the binomial coefficients. These numbers appear as the coefficients of terms in the expansion of the power of a binomial, and are also commonly found in combinatorics problems.

Question 18

Why is it called Pascal’s triangle?

Answer 18

Pascal’s triangle is named after Blaise Pascal, who published an important treatise on the subject in the 17th century. However, Pascal was not the discoverer of the triangle, it was already known in medieval Europe and much earlier in Asia.

Question 19

What is a sequence in math?

Answer 19

A sequence is a list of things, typically numbers. In a sequence, the order of the terms matters–that is, if you change order of the terms, then you get a different sequence.

Question 20

What is a famous mathematical sequence?

Answer 20

There are many famous sequences. Some of the most common are arithmetic sequences, geometric sequences, the Fibonacci sequence, the triangular number sequence, the square numbers sequence, and the cube numbers sequence.

Question 21

Arithmetic Sequence

Answer 21

is a sequence that increases or decreases by a constant amount with each number. The sequence from above, 5, 10, 15, 20, 25, … , is an example of an arithmetic sequence. Some other examples are shown below.

-5, -3, -1, 1, 3, …
6, 3, 0, -3, -6, …

Question 22

Geometric sequences

Answer 22

are sequences in which each number is multiplied (or divided) by the same value to get the next number. Whereas in arithmetic sequences, the operation is addition or subtraction, in geometric sequences, it is multiplication or division. The sequence is a geometric sequence that is generated by repeatedly dividing by 2 (or multiplying by ). Other examples of geometric sequences are shown below.

6, 12, 24, 48, 96, …
1, 1.1, 1.21, 1.331, 1.4641, …

Question 23

How do you write a conjecture?

Answer 23

To write a conjecture, first observe some information about the topic. After gathering some data, decide on a conjecture, which is something you think is true based on your observations.

Question 24

What is an example of a conjecture in math?

Answer 24

There are many conjectures in mathematics. Anything from finding the missing number in a sequence to Fermat’s Last Theorem, which is a theorem for which a full proof has not yet been found.

Question 25

How do you make a function table?

Answer 25

A traditional function table is made using two rows, with the top row being the input cells and bottom row being the output cells. Each column represents a single input/output relationship.

Question 26

What is a function rule for a table?

Answer 26

The rule of a function table is the mathematical operation that describes the relationship between the input and the output. The rule must be consistently applied to all input/output pairs.

Question 27

How do you know if a table is a function?

Answer 27

The table is a function if there is a single rule that can consistently be applied to the input to get the output. If the same rule doesn't apply to all input and output relationships, then it's not a function.