Skip to content | Accessibility - list of access keys used on this site | Sitemap
Find what you need to know

Home > Subjects > Mathematics > Level 1 > 1.3 Tables, equations and graphs (AS91028) > Subject content > Quadratic patterns

  • Subject: Mathematics
  • AS: 91028
  • Level: 1
  • Credits: 4
  • External

Mathematics 1.3 Investigate relationships between tables, equations and graphs

Quadratic patterns

e.g.  Find a rule for calculating the total number of boxes
        in each diagram, if the number of boxes in the
        bottom layer is known. 

algebraic diagram

        Draw up a table to show the number of boxes in the
        bottom layer and the total number of boxes in each
        diagram.

        Calculate the differences between the terms.

 

No. of boxes in bottom layer (n) 1 2 3 4 5  
Total no. of boxes (t) 1 3 6 10    
First difference                               2          3          4
Second difference                                1           1

 

Let the number of boxes in the bottom layer be n and the total number of boxes be t.

Because the second differences are the constant (they are all equal to 1), the rule in this example will be a quadratic expression.

The rule will be in the form: t = an² + bn + c – where a, b, and c are constants to be found.

Now when n=1, t=1 so 1 = a + b + c -------- (1)

When n=2, t=3 so 3 = 4a + 2b + c -------- (2)

When n=3, t=6 so 6 = 9a + 3b + c -------- (3)

Subtract equation (1) from equation (2) to eliminate c:
2 = 3a + b -------- (4)

Subtract equation (2) from equation (3) to eliminate c:
3 = 5a + b -------- (5)

This now gives two equations in a and b.
Solve these simultaneously to find a and b.

Subtract equation (4) from equation (5):
1 = 2a
a = (1/2)
Substitute this value for a in equation (4) to find b = (1/2)

Substitute these values for a and b into equation (1) to find c = 0.

Write a = algebraic formula, b = (1/2) and c = 0 into the rule.
Hence the rule for this pattern is t = (1/2)n² + (1/2)n

 

This method may be shortened by remembering that

a+b+c = first term

3a+b = first difference

2a = second difference        

 

For example for the pattern 10, 18, 28,.....

The equation goes an² + bn + c

 

10   18   28

   8    10

      2

 

a+b+c = 10  

3a+b = 8  

2a = 2

Therefore a = 1, b = 5 and c = 4. Equation is n²+5n+4

 

For more information see:
Finding the next number in a sequence
Quadratic sequences
Test bite

Back to top

0

People logged in

301

People surfing this site

Most recent posts

English Level 1
Length of Visual and Written essays

Posted by: englishteacher8

3:20pm 14.11.2017

English Level 2
91098 [2.1] Level 2 WRITTEN text post all your Qs here

Posted by: Aquiilla xo

5:57pm 20.11.2017

English Level 3 & Scholarship
Level 3 2017 Unfamiliar Texts- How Did It Go?

Posted by: Torpeylcr

8:47pm 14.11.2017

Mathematics Level 1
2017 LEVEL 1 MATHS EXAM COMMENTS - How did it go?

Posted by: littlefluffybear

6:00pm 21.11.2017

Mathematics Level 2
Probability and Simulation (91267, 91268)

Posted by: mathsteacher15

4:01pm 22.11.2017

Mathematics Level 3 Calculus & Calculus Scholarship
mathematic with calculus

Posted by: mathsteacher15

4:01pm 12.11.2017

Mathematics Level 3 Statistics & Statistics Scholarship
Level 3 Calculus Linear equations

Posted by: mathsteacher4

11:34am 13.11.2017

Sciences Level 1
SCIENCE 1.5 ACIDS & BASES 90944 EXTERNAL EXAM

Posted by: madeline5

2:29pm 05.11.2017

Physics Level 2
PHYSICS 2.6 ELECTRICITY AND ELECTROMAGNETISM

Posted by: scienceteacher3

11:42am 08.11.2017

Physics Level 3 & Scholarship
PHYSICS 3.1 PRACTICAL INVESTIGATION

Posted by: scienceteacher3

8:56pm 20.11.2017

Biology Level 2, 3 & Scholarship
NCEA Level 2 Cellular Respiration

Posted by: scienceteacher5

9:10pm 13.11.2017

Chemistry Level 2, 3 & Scholarship
CHEMISTRY 3.4 NEW Bonding and thermochemistry external

Posted by: scienceteacher13

10:16am 15.11.2017

Earth & Space Science
Where to find resources

Posted by: Josephbd

6:46pm 25.11.2015

Other
Who took year 11 general science?

Posted by: KingDavidFromFiji

9:03pm 30.10.2017