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Number Patterns - Grade 10

In earlier grades you saw patterns in the form of pictures and numbers. In this chapter, we learn more about the mathematics of patterns. Patterns are recognisable as repetitive sequences and can be found in nature, shapes, events, sets of numbers and almost everywhere you care to look. For example, seeds in a sunflower, snowflakes, geometric designs on quilts or tiles, the number sequence 0;4;8;12;16;...0;4;8;12;16;....

Investigation : Patterns

Can you spot any patterns in the following lists of numbers?

  1. 2;4;6;8;10;...2;4;6;8;10;...
  2. 1;2;4;7;11;...1;2;4;7;11;...
  3. 1;4;9;16;25;...1;4;9;16;25;...
  4. 5;10;20;40;80;...5;10;20;40;80;...

Common Number Patterns

Numbers can have interesting patterns. Here we list the most common patterns and how they are made.

Examples:

  1. 1;4;7;10;13;16;19;22;25;...1;4;7;10;13;16;19;22;25;... This sequence has a difference of 3 between each number. The pattern is continued by adding 3 to the last number each time.
  2. 3;8;13;18;23;28;33;38;...3;8;13;18;23;28;33;38;... This sequence has a difference of 5 between each number. The pattern is continued by adding 5 to the last number each time.
  3. 2;4;8;16;32;64;128;256;...2;4;8;16;32;64;128;256;... This sequence has a factor of 2 between each number. The pattern is continued by multiplying the last number by 2 each time.
  4. 3;9;27;81;243;729;2187;...3;9;27;81;243;729;2187;... This sequence has a factor of 3 between each number. The pattern is continued by multiplying the last number by 3 each time.

Special Sequences

Triangular Numbers

1 ; 3 ; 6 ; 10 ; 15 ; 21 ; 28 ; 36 ; 45 ; . . . 1 ; 3 ; 6 ; 10 ; 15 ; 21 ; 28 ; 36 ; 45 ; . . .

This sequence is generated from a pattern of dots which form a triangle. By adding another row of dots (with one more dot in each row than in the previous row) and counting all the dots, we can find the next number of the sequence.

Square Numbers

1 ; 4 ; 9 ; 16 ; 25 ; 36 ; 49 ; 64 ; 81 ; . . . 1 ; 4 ; 9 ; 16 ; 25 ; 36 ; 49 ; 64 ; 81 ; . . .

The next number is made by squaring the number of the position in the pattern. The second number is 2 squared (22or2×222or2×2

 
). The seventh number is 7 squared (72or7×772or7×7
 
) etc.

Cube Numbers

1 ; 8 ; 27 ; 64 ; 125 ; 216 ; 343 ; 512 ; 729 ; . . . 1 ; 8 ; 27 ; 64 ; 125 ; 216 ; 343 ; 512 ; 729 ; . . .

The next number is made by cubing the number of the position in the pattern. The second number is 2 cubed (23or2×2×223or2×2×2

 
). The seventh number is 7 cubed (73or7×7×773or7×7×7
  
) etc.

Fibonacci Numbers

0 ; 1 ; 1 ; 2 ; 3 ; 5 ; 8 ; 13 ; 21 ; 34 ; . . . 0 ; 1 ; 1 ; 2 ; 3 ; 5 ; 8 ; 13 ; 21 ; 34 ; . . .

The next number is found by adding the two numbers before it together. The 2 is found by adding the two numbers in front of it (1+11+1). The 21 is found by adding the two numbers in front of it (8+138+13). The next number in the sequence above would be 55 (21+3421+34).

Can you figure out the next few numbers?

Figure 1
Khan Academy video on Number Patterns - 1

Exercise 1: Study Table

Say you and 3 friends decide to study for Maths, and you are seated at a square table. A few minutes later, 2 other friends join you and would like to sit at your table and help you study. Naturally, you move another table and add it to the existing one. Now 6 of you sit at the table. Another 2 of your friends join your table, and you take a third table and add it to the existing tables. Now 8 of you can sit comfortably.

Figure 2: Two more people can be seated for each table added.
Figure 2 (MG10C7_002.png)

Examine how the number of people sitting is related to the number of tables.

Solution

  1. Step 1. Tabulate a few terms to see if there is a pattern :
    Table 1
    Number of Tables, nn Number of people seated
    1 4 = 4 4 = 4
    2 4 + 2 = 6 4 + 2 = 6
    3 4 + 2 + 2 = 8 4 + 2 + 2 = 8
    4 4 + 2 + 2 + 2 = 10 4 + 2 + 2 + 2 = 10
                   
    n n 4 + 2 + 2 + 2 + ... + 2 4 + 2 + 2 + 2 + ... + 2
  2. Step 2. Describe the pattern :

    We can see that for 3 tables we can seat 8 people, for 4 tables we can seat 10 people and so on. We started out with 4 people and added two each time. Thus, for each table added, the number of persons increased by 2.

Notation

Figure 3
Khan Academy video on Number Patterns

The nthnth-term of a sequence is written as anan. So for example, the 1st1st-term of a sequence is a1a1, the 10th10th-term is a10a10.

A sequence does not have to follow a pattern but when it does, we can often write down a formula to calculate the nthnth-term, anan. In the sequence

1 ; 4 ; 9 ; 16 ; 25 ; ... 1 ; 4 ; 9 ; 16 ; 25 ; ...
(1)

where the sequence consists of the squares of integers, the formula for the nthnth-term is

a n = n 2 a n = n 2
(2)

You can check this by looking at:

a 1 = 1 2 = 1 a 2 = 2 2 = 4 a 3 = 3 2 = 9 a 4 = 4 2 = 16 a 5 = 5 2 = 25 ... a 1 = 1 2 = 1 a 2 = 2 2 = 4 a 3 = 3 2 = 9 a 4 = 4 2 = 16 a 5 = 5 2 = 25 ...
(3)

Therefore, using Equation 2, we can generate a pattern, namely squares of integers.

We can also define the common difference for a pattern.

Definition 1: Common difference
The common difference is the difference between successive terms and is denoted by d.
For example, consider the sequence 10;7;4;1;...10;7;4;1;... . To find the common difference, we simply subtract each successive term:
7-10 = -3 4-7 = -3 1-4 = -3 7-10 = -3 4-7 = -3 1-4 = -3
(4)

Exercise 2: Study Table continued ....

As before, you and 3 friends are studying for Maths, and you are seated at a square table. A few minutes later, 2 other friends join you and add another table to the existing one. Now 6 of you can sit together. A short time later 2 more of your friends join your table, and you add a third table to the existing tables. Now 8 of you can sit comfortably as shown:

Figure 4: For each table added, two more people can be seated.
Figure 4 (MG10C7_003.png)

Find the expression for the number of people seated at nn tables. Then, use the general formula to determine how many people can sit around 12 tables and how many tables are needed for 20 people.

Solution

  1. Step 1. Tabulate a few terms to see if there is a pattern :
    Table 2
    Number of Tables, nn Number of people seated Formula
    1 4 = 4 4 = 4 = 4 + 2 · ( 0 ) = 4 + 2 · ( 0 )
    2 4 + 2 = 6 4 + 2 = 6 = 4 + 2 · ( 1 ) = 4 + 2 · ( 1 )
    3 4 + 2 + 2 = 8 4 + 2 + 2 = 8 = 4 + 2 · ( 2 ) = 4 + 2 · ( 2 )
    4 4 + 2 + 2 + 2 = 10 4 + 2 + 2 + 2 = 10 = 4 + 2 · ( 3 ) = 4 + 2 · ( 3 )
                   
    n n 4 + 2 + 2 + 2 + ... + 2 4 + 2 + 2 + 2 + ... + 2 = 4 + 2 · ( n - 1 ) = 4 + 2 · ( n - 1 )
  2. Step 2. Describe the pattern :

    The number of people seated at nn tables is:

    a n = a 1 + d · ( n - 1 ) a n = a 1 + d · ( n - 1 )
    (5)

    Notice how we have used d to represent the common difference. We could also have written a 2 in place of the d. We also used a 1 a 1 to represent the first term, rather than using the actual value (4).

  3. Step 3. Calculate the 12 th 12 th term :

    Considering the example from the previous section, how many people can sit around say 12 tables? We are looking for a12a12, that is, where n=12n=12:

    a n = a 1 + d · ( n - 1 ) a 12 = 4 + 2 · ( 12 - 1 ) = 4 + 2 ( 11 ) = 4 + 22 = 26 a n = a 1 + d · ( n - 1 ) a 12 = 4 + 2 · ( 12 - 1 ) = 4 + 2 ( 11 ) = 4 + 22 = 26
    (6)
  4. Step 4. Calculate the number of terms if an=20an=20 :
    a n = a 1 + d · ( n - 1 ) 20 = 4 + 2 · ( n - 1 ) 20 - 4 = 2 · ( n - 1 ) 16 ÷ 2 = n - 1 8 + 1 = n n = 9 a n = a 1 + d · ( n - 1 ) 20 = 4 + 2 · ( n - 1 ) 20 - 4 = 2 · ( n - 1 ) 16 ÷ 2 = n - 1 8 + 1 = n n = 9
    (7)
  5. Step 5. Final Answer :

    26 people can be seated at 12 tables and 9 tables are needed to seat 20 people.

It is also important to note the difference between nn and anan. nn can be compared to a place holder, while anan is the value at the place “held” by nn. Like our “Study Table” example above, the first table (Table 1) holds 4 people. Thus, at place n=1n=1, the value of a1=4a1=4 and so on:

Table 3
n n 1 2 3 4 ...
a n a n 4 6 8 10 ...

Investigation : General Formula

  1. Find the general formula for the following sequences and then find a10a10, a50a50 and a100a100:
    1. 2;5;8;11;14;...2;5;8;11;14;...
    2. 0;4;8;12;16;...0;4;8;12;16;...
    3. 2;-1;-4;-7;-10;...2;-1;-4;-7;-10;...
  2. The general term has been given for each sequence below. Work out the missing terms.
    1. 0;3;...;15;240;3;...;15;24       n2-1n2-1
    2. 3;2;1;0;...;-23;2;1;0;...;-2       -n+4-n+4
    3. -11;...;-7;...;-3-11;...;-7;...;-3       -13+2n-13+2n

Patterns and Conjecture

Figure 5
Khan Academy video on Number Patterns - 2

In mathematics, a conjecture is a mathematical statement which appears to be true, but has not been formally proven to be true. A conjecture can be seen as an educated guess or an idea about a pattern. A conjecture can be thought of as the mathematicians way of saying I believe that this is true, but I have no proof.

For example: Make a conjecture about the next number based on the pattern 2;6;11;17;...2;6;11;17;...

The numbers increase by 4, 5, and 6.

Conjecture: The next number will increase by 7. So, it will be 17+717+7 or 24.

Exercise 3: Number patterns

Consider the following pattern:

1 2 + 1 = 2 2 - 2 2 2 + 2 = 3 2 - 3 3 2 + 3 = 4 2 - 4 4 2 + 4 = 5 2 - 5 1 2 + 1 = 2 2 - 2 2 2 + 2 = 3 2 - 3 3 2 + 3 = 4 2 - 4 4 2 + 4 = 5 2 - 5
(8)
  1. Add another two rows to the end of the pattern.
  2. Make a conjecture about this pattern. Write your conjecture in words.
  3. Generalise your conjecture for this pattern (in other words, write your conjecture algebraically).
  4. Prove that your conjecture is true.
Solution
  1. Step 1. The next two rows :
    5 2 + 5 = 6 2 - 6 6 2 + 6 = 7 2 - 7 5 2 + 5 = 6 2 - 6 6 2 + 6 = 7 2 - 7
    (9)
  2. Step 2. Conjecture :

    Squaring a number and adding the same number gives the same result as squaring the next number and subtracting that number.

  3. Step 3. Generalise :

    We have chosen to use xx here. You could choose any letter to generalise the pattern.

    x 2 + x = ( x + 1 ) 2 - ( x + 1 ) x 2 + x = ( x + 1 ) 2 - ( x + 1 )
    (10)
  4. Step 4. Proof :
    Left side : x 2 + x Left side : x 2 + x
    (11)
    Right side : ( x + 1 ) 2 - ( x + 1 ) Right side : ( x + 1 ) 2 - ( x + 1 )
    (12)
    Right side = x 2 + 2 x + 1 - x - 1 = x 2 + x = Left side x 2 + x = ( x + 1 ) 2 - ( x + 1 ) Right side = x 2 + 2 x + 1 - x - 1 = x 2 + x = Left side x 2 + x = ( x + 1 ) 2 - ( x + 1 )
    (13)

Summary

  • There are several special sequences of numbers:
    • Triangular numbers 1 ; 3 ; 6 ; 10 ; 15 ; 21 ; 28 ; 36 ; 45 ; . . . 1 ; 3 ; 6 ; 10 ; 15 ; 21 ; 28 ; 36 ; 45 ; . . .
    • Square numbers 1 ; 4 ; 9 ; 16 ; 25 ; 36 ; 49 ; 64 ; 81 ; . . . 1 ; 4 ; 9 ; 16 ; 25 ; 36 ; 49 ; 64 ; 81 ; . . .
    • Cube numbers 1 ; 8 ; 27 ; 64 ; 125 ; 216 ; 343 ; 512 ; 729 ; . . . 1 ; 8 ; 27 ; 64 ; 125 ; 216 ; 343 ; 512 ; 729 ; . . .
    • Fibonacci numbers 0 ; 1 ; 1 ; 2 ; 3 ; 5 ; 8 ; 13 ; 21 ; 34 ; . . . 0 ; 1 ; 1 ; 2 ; 3 ; 5 ; 8 ; 13 ; 21 ; 34 ; . . .
  • We represent the nthnth-term with the notation anan
  • We can define the common difference of a sequence as the difference between successive terms.
  • We can work out a general formula for each number pattern and use that to predict what any number in the pattern will be.

Exercises

  1. Find the n th n th term for: 3,7,11,15,...3,7,11,15,... Click here for the solution
  2. Find the general term of the following sequences:
    1. -2;1;4;7;...-2;1;4;7;...
    2. 11;15;19;23;...11;15;19;23;...
    3. sequence with a3=7a3=7 and a8=15a8=15
    4. sequence with a4=-8a4=-8 and a10=10a10=10
    Click here for the solution
  3. The seating in a section of a sports stadium can be arranged so the first row has 15 seats, the second row has 19 seats, the third row has 23 seats and so on. Calculate how many seats are in the row 25. Click here for the solution
  4. A single square is made from 4 matchsticks. Two squares in a row need 7 matchsticks and 3 squares in a row need 10 matchsticks. Determine:
    1. the first term
    2. the common difference
    3. the formula for the general term
    4. how many matchsticks are in a row of 25 squares
    Figure 6
    Figure 6 (MG10C7_004.png)
    Click here for the solution
  5. You would like to start saving some money, but because you have never tried to save money before, you have decided to start slowly. At the end of the first week you deposit R5 into your bank account. Then at the end of the second week you deposit R10 into your bank account. At the end of the third week you deposit R15. After how many weeks do you deposit R50 into your bank account? Click here for the solution
  6. A horizontal line intersects a piece of string at four points and divides it into five parts, as shown below.
    Figure 7
    Figure 7 (MG10C7_005.png)
    If the piece of string is intersected in this way by 19 parallel lines, each of which intersects it at four points, find the number of parts into which the string will be divided. Click here for the solution

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