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Algebra : Key Stage 4 : Mathematics Content Knowledge


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Key Stage 4
Algebra
Question 2 of 34

2. How confident are you that you can find:

a. the nth term of an arithmetic sequence and the nth term of a geometric sequence?


Example

A sequence is a series of shapes or numbers which follow a rule.

An arithmetic sequence is a sequence of numbers with the rule ‘add a fixed number’. The number is the difference between consecutive terms.

For example, in the sequence 5, 8, 11, 14, 17, …, the rule is ‘add 3’. The nth term is 3n + 2. The 3 in 3n + 2 is the difference between consecutive terms. The +2 in 3n + 2 is the term that would come before the 1st term, sometimes called the zero term.

In general, for an arithmetic sequence with first term a and difference d, the nth term is given by:

Tn = a + (n – 1)d

A visual example of an arithmetic sequence:

Arithmetic sequence

Write down the number of dots in pattern 4. Find an expression, in terms of n, for the nth term in the sequence.
Does any pattern in the sequence contain 245 dots?

To create the next pattern requires adding 4 more dots, giving 19 dots, and the nth pattern will have 4n + 3 dots in it.
If a pattern has 245 dots then the equation 4n + 3 = 245 will have an integer solution. n =

245 – 3
4
 = 60.5 shows that no pattern in the sequence will have 245 dots in it.

A geometric sequence is a sequence of numbers with the rule ‘multiply by a fixed number’. The number, called the common ratio, is the ratio between successive terms.

For example, in the sequence 24, 36, 54, 81, 121.5, …, the rule is ‘multiply by 1.5’. The nth term is 24 × 1.5n-1.

In general, for a geometric sequence with first term a and common ratio r, the nth term is given by:

Tn = arn-1

In both cases the rule for finding the nth term involves applying the term-to-term rule exactly n – 1 times.

What this might look like in the classroom

Problem
A sequence starts with the first two terms 2 and 4. How could the sequence continue?

Possible response (there are plenty more though)
If we have the first two numbers in a sequence then it can be the beginning of an arithmetic sequence: in this case 2, 4, 6, 8 (add 2), or a
geometric sequence 2, 4, 8, 16 (multiply by 2)

The nth term of the arithmetic sequence is T(n) = 2n. It is quite easy to spot this just by looking (by inspection).This means that, for example, the 30th term will be 60.

The nth term of the geometric sequence is T(n) = 2n. The 30th term, for example, will be 1073741824.

Show that the difference between the 6th term of an arithmetic sequence starting 1, 5 and the 6th term of a geometric sequence starting 1, 5 is 3104

Taking this mathematics further

If the sequence is 2, 5, 8, 11
A finite series would be 2+5+8+11.
We could write the sum as

A sequence

We can add up these terms to get 26 but if there were many more terms we would need a formula.

If an arithmetic series has first term a and common difference d, then the sum of the first n terms of the progression is
1
2
n(2a + ( n − 1 )d).

Making connections

An understanding of sequences is an important step in generalising. Many using and applying problems can be generalised to an arithmetic sequence or a geometric sequence.

These basic sequences help to develop an understanding of the more complex sequences; e.g. consecutive squares found in A and AS level curriculum.

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