Igcse/ grade 9/ 10 Math 10-Geometric-Sequences.ppsx

Gcse sequeces

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apt4Maths
Minimising Workloads
Maximising Performance
GCSE (& Key Stage 3)
MATHEMATICS
Number; Algebra
Types of Number & Sequences
Geometric Sequences –Finding terms
and the nth term

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Lesson Objectives
•To investigate geometric sequences.
•To teach you how to find the next or
previous terms of a geometric sequence.
•To teach you how to find the expression for
the n
th
term of a geometric sequence.

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Types of Sequences
So far we have only learned about arithmeticsequences:
•Linear –when you add / subtractthe same amount on to the last
term each time.
•Quadratic –when you add / subtractthe same amount on to the
first difference each time.
•Cubic –when you add / subtractthe same amount on to the
second difference each time.
•Geometricsequences are when you are multiplyingeach term
by the same amount each term.
•Fibonacci sequences are where previous terms are added
togetherto get the next term.

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Examples
Arithmeticseq:6 13 20 27 34…
Common difference: +7 +7 +7 +7…
Geometricseq:2 6 18 54 162…
Common ratio: ×3 ×3 ×3 ×3…
Fibonacci seq:1 3 4 7 11…
(1+3) (3+4) (4+7)
Rule: Add the last two terms together…

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Finding Terms of Geometric Sequences
•Find the common ratio (iethe value you are multiplying
by) by dividing any two consecutive terms –the later one
by the earlier one.
•If you don’t have consecutive terms, divide one term by
an earlier term and then find the square root(if they are
two numbers apart) or cube root(if they are three
numbers apart) to find the common ratio.
•For the next term, multiplythe last term by the common ratio.
•For an earlier term, dividethe term given by the common ratio.

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Example 1
Find the next and missing term of this sequence:
20000 4000 .…..160 32…
•Ratio: ×0.2 ×0.2…
•Common ratio = ×0.2 (which is the same as ÷5)
•Next term = 32 ×0.2= 6.4
•Missing term is either…
4000 ×0.2= 800(Going forward apply ratio)
or160 ÷0.2= 800(Going backward apply inverse)

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Example 2
Find the next and missing terms of this sequence:
7 .…..28 ……112
•Ratio ×Ratio:×4
•Common ratio = √4= ×2
•Next term = 112 ×2= 228
•Second term = 7 ×2= 14 (or 28 ÷2= 14)
•Fourth term = 28 ×2= 56 (or 112 ÷2= 56)

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Practice
Find the missing terms in these geometric sequences:
1) 4 20 100500___
2) ___5 10 ___40
3) 4000400___ 4___
When you have worked out the answers, click on to the next slide to check them.

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Answers
1) 4 20100500
×5×5 ×5
r = 20 ÷4 = 5
2) ___ 510 ___40
×2
r = 10 ÷5 = 2
3) 4000400___ 4___
×0.1
r = 400 ÷4000 = 0.1
2500
2.5 20
40 0.4
×5
×2÷2
×0.1×0.1
×2
×0.1

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Finding the n
th
Term –Geometric
To find the n
th
term of a GEOMETRIC sequence…
Use the general formula:ar
(n –1)
or a ×r
(n –1)
Where… a = first term and r = common ratio
Eg 3 6 1224…
a = 3
r = ×2
Expression for the n
th
term = 3 ×2
(n –1)
Checkby substituting n=4, evaluating and checking that the
answer gives the 4
th
term value of 24…
3 ×2
(4–1)
3 ×2
3
3 ×8 = 24

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Practice
Find the expression for the n
th
term of these sequences:
1) 5 20 80 320…
2) ? 60 30 15…
3) 1 ? 1 1
16 ? 4 2…
When you have worked out the answers, click on to the next slide to check them.

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Answers
1)5 2080320…
a = 5 r = 20÷5= 4
n
th
term: 5 ×4
(n-1)
2)? 603015…
r = 30÷60= 0.5a = 60÷0.5 = 120
n
th
term: 120 ×0.5
(n-1)
3) 1 ? 1 1
16 ? 4 2…
a =
1
/
16 r =
1
/
2÷
1
/
4= 2
n
th
term:
1
/
16×2
(n-1)
Check 3rd value:
5 ×4
(3-1)
= 5 ×4² = 80
Check 3rd value:
120 ×0.5
(3-1)
= 120 ×0.5² = 30
Check 3rd value:
1
/
16×2
(3-1)
=
1
/
16×2² =
1
/
4

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What next?
•It would be wise to make some notes and write down
some examples.
•We haven’t really done many practice questions, and
so you may like to do a few more -using online
resources or past paper questions.
•You could now put together an A4 revision sheet on
the first half of this module (based on types of
number and number sequences) and then move on
to the work on co-ordinates and graphs.