## Key Concepts

- Numerical sequences
- Addition sequences
- Multiplication sequences
- Numerical patterns from the word problems

### Introduction

In this chapter, we will learn about numerical sequences, addition sequences, multiplication sequences, and word problems.

## Numerical sequences

**Numerical sequence is a list of numbers arranged by a pattern or rule.**

The pattern or rule of a sequence can be adding a number or subtracting a number.

Numerical sequence is one of the best concepts for the students to sharpen their numerical reasoning capability.

**Example:** Find the missing number

4, 11, 18, 25, _____.

**Solution:** In this example, the difference between each number is 7.

The rule for this sequence is to add 7 each time.

The missing number in the sequence: 25 + 7 = 32

### Addition sequences

Addition sequence is a sequence in which a particular number is added each time.

**Example 1: **

A shopkeeper sells balls. His sales keep increasing by 2 balls per day.

Days | Balls |

1 | 1 |

2 | 3 |

3 | 5 |

4 | 7 |

Using the numbers, the sequence is:

The rule to go from one number to the next number is ‘add 2’.

**Example 2:**

Maria has $5. She will save $10 each week. Steepen has $8 and will also save $10 each week.

Maria uses the rule “add 10” to create tables to see how much each will have saved after each week.

What relationship do you notice between the corresponding terms?

At the end of each week, Steepen has saved $3 more than Maria.

**Example 3:**

**Patterns on the Calendar**: Possible increasing patterns include

- In each row numbers increase by 1.
- In each column numbers increase by 7.
- On the diagonal from left to right, numbers increase by 8.
- On the diagonal from right to left, numbers increase by 6.

### Multiplication sequences

A sequence made by multiplying by the same value each time.

**Example 1**: 2, 4, 8, 16, 32, 64, 128, 256, …

(Each number is 2 times the number before it)

**Example** **2**: 3, 6, 12, 24, 48, …

(Each number is 2 times the number before it)

This below image indicates multiplication sequence

**Example 3:**

Jack is preparing for an exam. Each week he reads 20 pages in text book, 40 pages in work book.

He created a table to record his reading. Find the total number of pages that he will read from the text book and the work book after 5 weeks? Can you identify any relationship between reading the text book and the work book?

**Solution:**

Since jack reads 20 pages each week. Add 20 to find the next term for the total pages he reads.

And add 40 to find each term in the pattern for the total number of pages he reads.

Compare the corresponding terms in the patterns:

20×2 = 40

40×2 = 80

60×2 = 120

80×2= 160

100×2 = 200

So, the total number of work book pages is always 2 times the total number of text book pages.

**Example 4:**

Sheila and Patrick are making a table to compare gallons, quarts, and pints. Use the rule “add 4” to complete the column for the number of quarts. Then, use the rule “add 8” to complete the column for the number of pints.

**Solution:**

Compare the corresponding terms in the patterns:

4×2 = 8

8×2 = 16

12×2 = 24

16×2= 32

20×2 = 40

24×2= 48

So, the total number of pints is always 2 times the total number of quarts.

## Exercise

- Consider the following sequence. What number should come next?
- 1, 3, 5, 7…. b) 3, 6, 11, 18……
- Consider the following sequence. What is the missing number?
- 26, 21, …, 11, 6. b) 19, …, 31, 37
- Find the missing terms in the following sequence:

8, _, 16, _, 24, 28, 32

- If hickory tree continues to grow 1 feet each year, how tall will the tree be after 10 years?
- Riya’s weight increases by 5 kgs every year. What will be Riya’s weight after 5 years?
- Find the next number in the sequence:
- 4, 8, 16, 32, ___ b) 5, 10, 20, 40, __
- What should come next:
- 2, 4, 9, 11, 16, … b) 30, 28, 25, 21, 16, …
- Each week Stem lifts weights twice and runs 4 times. Stem uses the rules “add 2” and “add 4” to complete the table. What relationship do you notice between the corresponding terms?
- Bridget is organizing her books and putting them on shelves. She put 72 books on the first shelf, 81 books on the second shelf, 90 books on the third shelf, and 99 books on the fourth shelf. If this pattern continues, how many books will Bridget put on the fifth shelf?
- Marcy is sorting pencils into boxes. She put 2 pencils in the first box, 4 pencils in the second box, 8 pencils in the third box, and 16 pencils in the fourth box. If this pattern continues, how many pencils will Marcy put in the fifth box?

### What have we learned

- Numerical sequences
- Addition sequences
- Multiplication sequences
- Numerical patterns from the word problems