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This lesson submitted by amanda.bragg

**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:

*a*2−*a*1=2−1 =1

*a*3−*a*2=4−2 =2

*a*4−*a*3=7−4 =3

*a*5−*a*4 =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 *n*th term of the sequence 5,12,23,38,...,...,

**General Case**

If the sequence is quadratic, the *n*th term should be *Tn*=*an*2+*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 2*a*. 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.

**Derivation of the ***n*th**-term of a Quadratic Sequence**

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

*an*=*A*⋅*n*2+*B*⋅*n*+*C*

where *A*, *B* and *C* are some constants to be determined.

*An*=* A*⋅*n*2+*B*⋅*n*+*C*

*a*1= A(1)2+*B*(1)+*C*=*A*+*B*+*C*

*a*2= A(2)2+*B*(2)+*C*=4*A*+2*B*+*C*

*a*3= A(3)2+*B*(3)+*C*=9*A*+3*B*+*C*

Let*d*∴≡* a*2−*a*1

*d*= 3A+*B*

⇒*B*=*d*−3*A*

The common second difference is obtained from

*D*= (*a*3−*a*2)−(*a*2−*a*1)

= (5*A*+*B*)−(3*A*+*B*)

= 2*A*

⇒*A*=*D*2

Therefore, from **Equation 10**,

*B*=*d*−32⋅*D*

From **Equation 8**,

*C*=*a*1−(*A*+*B*)=*a*1−*D*2−*d*+32⋅*D*

∴*C*=*a*1+*D*−*d*

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

*an*=*D*2⋅*n*2+(*d*−32*D*)⋅*n*+(*a*1−*d*+*D*)

**EXERCISE 3: Using a set of equations**

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

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

**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**

- Find the first 5 terms of the quadratic sequence defined by:

*an*=*n*2+2*n*+1

**(20)**

- Determine which of the following sequences is a quadratic sequence by calculating the common second difference:
- 6;9;14;21;30;...
- 1;7;17;31;49;...
- 8;17;32;53;80;...
- 9;26;51;84;125;...
- 2;20;50;92;146;...
- 5;19;41;71;109;...
- 2;6;10;14;18;...
- 3;9;15;21;27;...
- 10;24;44;70;102;...
- 1;2,5;5;8,5;13;...
- 2,5;6;10,5;16;22,5;...
- 0,5;9;20,5;35;52,5;...
- Given
*an*=2*n*2, find for which value of*n*,*an*=242 - Given
*an*=(*n*−4)2, find for which value of*n*,*an*=36 - Given
*an*=*n*2+4, find for which value of*n*,*an*=85 - Given
*an*=3*n*2, find*a*11 - Given
*an*=7*n*2+4*n*, find*a*9 - Given
*an*=4*n*2+3*n*−1, find*a*5 - Given
*an*=1,5*n*2, find*a*10 - For each of the quadratic sequences, find the common second difference, the formula for the general term and then use the formula to find
*a*100. - 4,7,12,19,28,...
- 2,8,18,32,50,...
- 7,13,23,37,55,...
- 5,14,29,50,77,...
- 7,22,47,82,127,...
- 3,10,21,36,55,...
- 3,7,13,21,31,...
- 3,9,17,27,39,...

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