C2: Geometric Series

Dr J Frost (jfrost@tiffin.kingston.sch.uk)

Last modified: 24th September 2013

Types of series

2, 5, 8, 11, 14, …+3 +3 +3 This is a:

Arithmetic Series?

Geometric Series?3, 6, 12, 24, 48, …×2 ×2 ×2

common difference

common ratio

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Common Ratio

Identify the common ratio :

1 ,2 ,4 ,8 ,16 ,32 ,… 𝑟=21

24 ,18 ,12 ,8 ,… 𝑟=2/32

10 ,5 ,2.5 ,1.25 ,… 𝑟=1/23

5 ,−5 ,5 ,−5 ,5 ,−5 ,… 𝑟=−14

𝑥 ,−2 𝑥2 , 4 𝑥3 𝑟=−2 𝑥5

1 ,𝑝 ,𝑝2 ,𝑝3 ,… 𝑟=𝑝6

4 ,−1 ,0.25 ,−0.0625 ,… 𝑟=−0.257

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Common Ratio Exam QuestionMay 2013 (Retracted)

Hint for (a): the common ratio between the first and second terms, and the second and third terms, is the same.

𝑟=3𝑝+154𝑝 =

3020=

32

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b

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th term

Arithmetic Series Geometric Series

𝑈𝑛=𝑎+(𝑛−1 )𝑑 𝑈𝑛=𝑎𝑟 𝑛−1

Determine the following:

3, 6, 12, 24, … 𝑈 10=1536

40, -20, 10, -5, … 𝑈 10=−564

𝑈𝑛=(−1 )𝑛− 1× 52𝑛− 4

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Another Common Ratio Example

The numbers and form the first three terms of a positive geometric sequence. Find:

a) The possible values of .b) The 10th term in the sequence.

But there are no negative terms so

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Missing information

The second term of a geometric sequence is 4 and the 4th term is 8.The common ratio is positive. Find the exact values of:a) The common ratio.b) The first term.c) The 10th term.

a) Dividing (1) by (2) gives b) Substituting, c)

Bro Tip: Explicitly writing first helps you avoid confusing the th term with the ‘sum of the first terms’ (the latter of which we’ll get onto).

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th term with inequalities

What is the first term in the geometric progression to exceed 1 million?

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Exam QuestionEdexcel June 2010

25000×1.03=25750

𝑟=1.03

𝑈 𝑁>40000

𝑆10=25000 (1−1.0310 )

1−1.03=£ 287,000

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Sum of the first terms

Arithmetic Series Geometric Series

𝑆𝑛=𝑛2 (2𝑎+ (𝑛−1 )𝑑 ) 𝑆𝑛=

𝑎 (1−𝑟𝑛)1−𝑟

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Technically you could be asked in an exam the proof of the sum of a geometric series (it once came up!)So let’s prove it…

Sum of the first terms

Geometric Series 𝑆𝑛=𝑎 (1−𝑟𝑛)1−𝑟

3 ,6 ,12 ,24 , 48 ,…

Find the sum of the first 10 terms.

𝑎=3 ,𝑟=2 ,𝑛=104 ,2,1 , 12 ,

14 ,18 ,… 𝑎=4 ,𝑟=

12 ,𝑛=10

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Summation Notation

Find

𝑎=6 ,𝑟=2 ,𝑛=10

𝑆10=6 (1−210 )1−2

=6138

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Harder Questions: Type 1

Find the least value of such that the sum of to terms would exceed 2 000 000.

An investor invests £2000 on January 1st every year in a savings account that guarantees him 4% per annum for life. If interest is calculated on the 31st of December each year, how much will be in the account at the end of the 10th year?

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Exercise 7DFind the sum of the following geometric series (to 3dp if necessary).

a) (8 terms)c) (6 terms)e) h)

The sum of the first three terms of a geometric series is 30.5. If the first term is 8, find the possible values of .

Jane invest £4000 at the start of every year. She negotiates a rate of interest of 4% per annum, which is paid at the end of the year. How much is her investment worth at the end of (a) the 10th year and (b) the 20th year. (a) (b)

A ball is dropped from a height of 10m. It bounces to a height of 7m and continues to bounce. Subsequent heights to which it bounces follow a geometric sequence. Find out:a) How high it will bounce after the fourth bounce,b) The total distance travelled until it hits the ground for a sixth time.

Find the least value of such that the sum to terms would first exceed 1.5 million.

1

2

4

5

6

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Different types of series

1 + 2 + 4 + 8 + 16 + ...

What can you say about the sum of each series up to infinity?

1 + 2 + 3 + 4 + 5 + ...

1 + 0.5 + 0.25 + 0.125 + ...

11+12 +13+14 +…

This is divergent – the sum of the values tends towards infinity.

This is divergent – the sum of the values tends towards infinity. But arguably, the sum of the natural numbers is .

This is convergent – the sum of the values tends towards a fixed value, in this case 2.

This is divergent . This is known as the Harmonic Series

11+14 +19+

116 +…

This is convergent . This is known as the Basel Problem, and the value is .

Just for fun...

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Sum to InfinityThink about our formula for the sum of the first terms. If we make infinity, what do we require of for not to be infinity (i.e. we want to keep the series convergent). And what will the formula become?

Restriction on : ?

𝑆𝑛=𝑎 (1−𝑟𝑛)1−𝑟

𝑆∞=𝑎1−𝑟?

Examples

1 , 12 ,14 ,18 ,…

𝒂=𝟏 ,𝒓=𝟏𝟐 𝑺∞=𝟐

27 ,−9,3 ,−1 ,… 𝒂=𝟐𝟕 ,𝒓=−𝟏𝟑 𝑺∞=𝟖𝟏𝟒

𝑝 ,𝑝2 ,𝑝3 ,𝑝4 ,… 𝒂=𝒑 ,𝒓=𝒑𝑺∞=𝒑

𝟏−𝒑h𝑤 𝑒𝑟𝑒−1<𝑝<1

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𝑝 ,1 , 1𝑝 ,1𝑝2,… 𝒂=𝒑 ,𝒓=

𝟏𝒑 𝑺∞=

𝒑𝟐

𝒑−𝟏???

A somewhat esoteric Futurama joke explained

Bender (the robot) manages to self-clone himself, where some excess is required to produce the duplicates (e.g. alcohol), but the duplicates are smaller versions of himself. These smaller clones also have the capacity to clone themselves. The Professor is worried that the total amount mass consumed by the growing population is divergent, and hence they’ll consume to Earth’s entire resources.

A somewhat esoteric Futurama joke explained

This simplifies to

The sum is known as the harmonic series, which is divergent.

Another Example

The sum to 4 terms of a geometric series is 15 and the sum to infinity is 16.a) Find the possible values of .

b) Given that the terms are all positive, find the first term in the series.

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Another Example

Edexcel May 2011

𝒓=𝟑𝟒

𝒂=𝟐𝟓𝟔𝑺∞=𝟏𝟎𝟐𝟒

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Exercises

Exercise 7D Q6, 7Exercise 7E Q8Exercise 7F Q10