Quadratic Sequences

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This lesson was written by 
Rory Adams
Objectives 
Students will find terms in a quadratic sequence.
Students will find differences.
Procedure 

Introduction

In Grade 10, you learned about arithmetic sequences, where the difference between consecutive terms was constant. In this chapter we learn about quadratic sequences.

What is a quadratic sequence?

 

DEFINITION 1: Quadratic Sequence

A quadratic sequence is a sequence of numbers in which the second differences between each consecutive term differ by the same amount, called a common second difference.

For example,

1;2;4;7;11;...

 

is a quadratic sequence. Let us see why ...

If we take the difference between consecutive terms, then:

a2−a1=2−1 =1

a3−a2=4−2 =2

a4−a3=7−4 =3

a5−a4 =11−7=4

 

We then work out the second differences, which is simply obtained by taking the difference between the consecutive differences {1;2;3;4;...} obtained above:

2−1 = 1

3−2 = 1

4−3... = 1

 

We then see that the second differences are equal to 1. Thus, Equation 1 is a quadratic sequence.

Note that the differences between consecutive terms (that is, the first differences) of a quadratic sequence form a sequence where there is a constant difference between consecutive terms. In the above example, the sequence of {1;2;3;4;...}, which is formed by taking the differences between consecutive terms of Equation 1, has a linear formula of the kind ax+b.

 

Quadratic Sequences

The following are also examples of quadratic sequences:

3;6;10;15;21;...

4;9;16;25;36;...

7;17;31;49;71;...

2;10;26;50;82;...

31;30;27;22;15;...

 

Can you calculate the common second difference for each of the above examples?

EXERCISE 1: Quadratic sequence

Write down the next two terms and find a formula for the nth term of the sequence 5,12,23,38,...,...,

[ SHOW SOLUTION ]

General Case

If the sequence is quadratic, the nth term should be Tn=an2+bn+c

 

TERMS           a+b+c             4a+2b+c                     9a+3b+c         

1st difference             3a+b               5a+b               7a+b

2nd difference                                   2a                   2a       

 

In each case, the second difference is 2a. This fact can be used to find a, then b then c.

 

EXERCISE 2: Quadratic Sequence

The following sequence is quadratic: 8,22,42,68,... Find the rule.

[ SHOW SOLUTION ]

 

Derivation of the nth-term of a Quadratic Sequence

Let the nth-term for a quadratic sequence be given by

an=An2+Bn+C

 

where AB and C are some constants to be determined.

An= An2+Bn+C

a1= A(1)2+B(1)+C=A+B+C

a2= A(2)2+B(2)+C=4A+2B+C

a3= A(3)2+B(3)+C=9A+3B+C

 

Letd∴≡ a2−a1

d= 3A+B

B=d−3A

 

The common second difference is obtained from

D= (a3−a2)−(a2−a1)

= (5A+B)−(3A+B)

= 2A

 

A=D2

 

Therefore, from Equation 10,

B=d−32⋅D

 

From Equation 8,

C=a1−(A+B)=a1−D2−d+32⋅D

 

C=a1+Dd

 

Finally, the general equation for the nth-term of a quadratic sequence is given by

an=D2⋅n2+(d−32D)⋅n+(a1−d+D)

 

EXERCISE 3: Using a set of equations

Study the following pattern: 1; 7; 19; 37; 61; ...

  1. What is the next number in the sequence ?
  2. Use variables to write an algebraic statement to generalise the pattern.
  3. What will the 100th term of the sequence be ?

[ SHOW SOLUTION ]

 

Plotting a graph of terms of a quadratic sequence

Plotting an vs. n for a quadratic sequence yields a parabolic graph.

Given the quadratic sequence,

3;6;10;15;21;...

 

If we plot each of the terms vs. the corresponding index, we obtain a graph of a parabola.

 

http://cnx.org/content/m32633/latest/MG11C5_001.png (Figure One)

 

End of chapter Exercises

  1. Find the first 5 terms of the quadratic sequence defined by:

an=n2+2n+1

(20)

  1. Determine which of the following sequences is a quadratic sequence by calculating the common second difference:
    1. 6;9;14;21;30;...
    2. 1;7;17;31;49;...
    3. 8;17;32;53;80;...
    4. 9;26;51;84;125;...
    5. 2;20;50;92;146;...
    6. 5;19;41;71;109;...
    7. 2;6;10;14;18;...
    8. 3;9;15;21;27;...
    9. 10;24;44;70;102;...
    10. 1;2,5;5;8,5;13;...
    11. 2,5;6;10,5;16;22,5;...
    12. 0,5;9;20,5;35;52,5;...
  2. Given an=2n2, find for which value of nan=242
  3. Given an=(n−4)2, find for which value of nan=36
  4. Given an=n2+4, find for which value of nan=85
  5. Given an=3n2, find a11
  6. Given an=7n2+4n, find a9
  7. Given an=4n2+3n−1, find a5
  8. Given an=1,5n2, find a10
  9. For each of the quadratic sequences, find the common second difference, the formula for the general term and then use the formula to find a100.
    1. 4,7,12,19,28,...
    2. 2,8,18,32,50,...
    3. 7,13,23,37,55,...
    4. 5,14,29,50,77,...
    5. 7,22,47,82,127,...
    6. 3,10,21,36,55,...
    7. 3,7,13,21,31,...
    8. 3,9,17,27,39,...

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