# StudySquare

This page covers the following topics:

1. Generating terms

2. Triangular sequences

3. Fibonacci sequence

A sequence is an ordered list of numbers that can be described by a specific rule. Each number in a sequence is called a term of the sequence and each term takes a position in it. The term to term rule of a sequence describes how to get from one term to the next in the sequence by looking for a pattern in moving between consecutive terms.

An increasing sequence is made up of increasing terms, whereas a decreasing sequence has decreasing terms. A periodic sequence is one whose terms repeat at regular intervals. A finite sequence has a finite number of terms, whereas an infinite sequence goes on forever.

The triangular sequence has first term of 1, second term of 1 + 2, third term of 1 + 2 + 3, and so on. This is called a triangular sequence, because one can represent the sequence as a triangle of dots. A term of the sequence can be generated if the 2 previous terms are known. The nth term of the triangular sequence is the sum of the first n natural numbers 1 + … + n = n(n + 1)/2.

The Fibonacci sequence is an infinite sequence given by {1, 1, 2, 3, …}. The first 2 terms are 1 and 1, and each next term is produced by adding the previous 2 terms together. Choosing different values for the first 2 terms, but following the rule of adding the 2 previous terms to get the next term, gives different Fibonacci-type sequences.

Placing squares of length corresponding to a Fibonacci number next to each other produces a spiral through the squares as shown. This is the Fibonacci Spiral, which is a very significant pattern that appears in nature. The nth term of the Fibonacci sequence is the sum of term (n − 1) and term (n − 2), provided n is greater than 2.

# 1

Given the Fibonacci sequence {1, 1, 2, 3, 5, 8, …}, what are the next 2 terms?

Each next term of the Fibonacci sequence is produced by adding the previous 2 terms together.

8 + 5 = 13

13 + 8 = 21

13, 21

# 2

Amie is training as a goalie for his school team. With every training session, she notices that he is saving three more goals than she did during the last session. What is the term to term rule of the sequence of goals saved?

The sequence of the goals saved by Amie is {1, 4, 7, ...}.

The term to term rule is that the number of goals saved increases by 3 compared to the last session.

Increase previous number by 3.

# 3

Why does the order of numbers in a sequence matter?

The order in a sequence matters as the differences between terms will determine the nth term expression of the sequence.

Differences between terms matter.

# 4

What are the first 3 terms of the triangular sequence?

1st term: 1

2nd term: 1 + 2 = 3

3rd term: 1 + 2 + 3 = 6

1, 3, 6

# 5

Adam notices he ate 2 apples two weeks ago, 4 apples the week after and 6 apples this week. What is the 2nd term of the sequence of apples?

The sequence of apples eaten by Adam is given by {2, 4, 6}.

Thus, the second term is 4.

4

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