Quadratic Sequences (Cambridge International IGCSE Mathematics)

Quadratic Sequences

What is a quadratic sequence?

• A quadratic sequence has an **n th term formula that involves n²

• The second differences are** constant (the same)

  • These are the differences between the first differences
  • For example, 3, 9, 19, 33, 51, … 1st Differences: 6, 10, 14, 18, ... 2nd Differences: 4, 4, 4, ...

How do I find the n^th term formula for a simple quadratic sequence?

• The sequence with the nth term formula n² are the square numbers

  • 1, 4, 9, 16, 25, 36, 49, ...

    • From 1², 2², 3², 4², ...

• Some sequences are just the square numbers but with a different starting number

  • 16, 25, 36, 49, ... has the formula (n + 3)²

    • Substitute in n = 1, n = 2, n = 3 to see why

  • 0, 1, 4, 9, ... has the formula (n - 1)²

• Finding the nth term formula comes from comparing sequences to the square numbers

  • 2, 5, 10, 17, 26, 37, 50, ... has the formula n² + 1

    • Each term is 1 more than the square numbers

  • 2, 8, 18, 32, 50, 72, 98, ... has the formula 2n²

    • Each term is double a square number

• It may be a simple combination of the above

  • For example, doubling then adding 1

    • 3, 9, 19, 33, 51, 73, 99, ... has the formula 2n² + 1

  • For example, subtracting 2 from square numbers starting with 4

    • 2, 7, 14, 23, ... has the formula (n + 1)² - 2

How do I find the n^th term formula for any quadratic sequence?

• STEP 1 Work out the sequences of first and second differences

  • e.g. for the sequence 1, 10, 23, 40, 61

• STEP 2 Divide the second difference by 2 to find the coefficient of n²

  • e.g. a = 4 ÷ 2 = 2

• STEP 3 Write out the first three or four terms of an² and subtract the terms from the corresponding terms of the given sequence

  • e.g. for the sequence 1, 10, 23, 40, 61

• STEP 4 Work out the nth term of these differences to find the bn + c part of the formula

  • e.g. the nth term of -1, 2, 5, 8, ... is bn + c = 3n − 4

• STEP 5 Find an²* *+ bn + c by adding together this linear nth term to an² term

  • e.g. an²* + bn + c = 2n*² + 3n − 4

Exam Hint You must learn the square numbers from 1² to 15² .

Worked Example For the sequence 6, 9, 14, 21, 30, ....

(a) Find a formula for the n^th term.

Answer:

Find the first and second differences

Sequence: 6, 9, 14, 21, 30

• First differences:   3,    5,     7,     9, ...           *
  

• Second differences: 2, 2, 2, ... *

*Halve the second difference, this is the value of a * *Write down the sequence *an² Compare this to the original sequence

• Sequence: 6, 9, 14, 21, 30, ..*.
  1n² **: 1, 4, 9, 16, 25, ... * Differences: 5, 5, 5, 5, 5, ...

*Each term is 5 more than the terms in n², so add 5 to *n²

Final answer: n^th** term =** n ² + 5 **

(b) Hence, find the 20^th term of the sequence.

Answer:

*Substitute n = 20 into *n² + 5

(20)² + 5 = 400 + 5

Final answer: The 20^thterm is 405

Worked Example For the sequence 5, 7, 11, 17, 25, ....

(a) Find a formula for the n^th term.

Answer: Start by finding the first and second differences

Sequence: 5, 7, 11, 17, 25

First differences: 2, 4, 6, 8, ...

Second difference: 2, 2, 2, ...

Hence

a = 2 ÷ 2 = 1*

Now write down an² (just n² in this case as a = 1) and subtract the terms from the original sequence

sequence: 5, 7, 11, 17, ...

an^2. **: 1, 4, 9, 16, ...*

• difference: 4, 3, 2, 1, ... *

*Work out the nth term of these differences to give you bn + *c

bn + c = −n + 5*

*Add an² and bn + **c **together to give you the *n^th term of the sequence

Final answer: n^th** term =** n**²** − n + 5**

(b) Hence find the 20^th term of the sequence.

Answer:

Substitute n = 20 into n² − n + 5

*(20)² − 20 + 5 = 400 − 15 *

Final answer: 20^thterm = 385