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Sequences

A sequence is a list of numbers. A sequence has terms and values. The term number is often written as \(n\).

Example

Tom, Mary, Jane and Jack are all standing in a queue. Tom is at rhe front of the queue.

nth term derivation

Since Tom is first , he is the first term , n= 1

nth term derivation

The value of \(n = 1\) is Tom. The value of \(n = 3\) is Jane. Jack is the fourth term.

Example

3, 6, 9, 12, 15, 18 , 21,...

The value of \(n = 1\) is 3 , so 3 is the first term.

The value of \(n = 4\) is 12 , so 12 is the fourth term.

The number 15 appears fifth in the list, so is term \(n = 5\) .

To find the rule of a sequence

  1. Write out the sequence.
  2. Write the term numbers underneath.
  3. Find the difference between the values.
  4. If the difference is constant, multiply \(n\) by this number.
  5. Leave a gap, draw a line, copy the sequence again.
  6. Find the constant needed to match the sequence values.
  7. Write the expression for the \(n\)th term.
Example

Find the rule for the sequence: 5, 8, 11, 14…

\[ \text{difference} = 3 \]
nth term derivation

The rule is \(3n + 2\).

Example

Given the sequence 1, 4, 7, 10…

    a) Find the next three numbers  in the sequence. 
    b) What is the rule for the nth term of this sequence ?
    c) Find the value of the 20th term.
    d) What term has a value of 61 ? 

a) Next three terms: 13, 16, 19

b) Rule for the nth term:

\[ \text{difference} = 3 \]
nth term derivation
\[ \text{rule} = 3n - 2 \]

c) Value of the 20th term:

\[ 3(20) - 2 = 58 \]

d) Which term has value 61?

\[ 3n - 2 = 61 \] \[ 3n = 63 \] \[ n = 21 \]

Picture Sequences

Example

Find the rule connecting bars \(b\) to dots \(d\):

picture sequence 1 picture sequence 2 picture sequence 3

Each time a dot is added, 2 bars are added.

Write the term number above the pictures:

term numbers above pictures

n has been replaced by d

Proceed as before:

bars and dots rule derivation
\[ \text{rule is } b = 2d - 1 \]

To find the number of bars, double the number of dots and subtract 1

How many bars for 52 dots?

\[ b = 2d - 1 \] \[ b = 2(52) - 1 = 103 \]

How many dots for 51 bars?

\[ 51 = 2d - 1 \] \[ 2d = 52 \] \[ d = 26 \]

Non-constant Difference

If the difference between values is not constant, look for a constant between the differences.

The nth term will contain \(an^2\), where \(a = \frac{k}{2}\).

Square each term, then multiple by ½k

Example

Find the rule for the sequence: 6, 18, 38, 66…

quadratic sequence derivation
\[ \text{rule} = 4n^2 + 2 \]

Check for \(n = 2\):

\[ 4n^2 + 2 = 4(2)^2 + 2 = 16 + 2 = 18 \]
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