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32: Geometric Sequences and Series Part 1 © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules

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Geometric Sequences and Series Module C2 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

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Geometric Sequences and Series There is a legend that Zarathustra, a Persian, invented chess to give interest to the life of the king who was bored. For his reward, Zarathustra asked for a quantity of grain, according to the following rules.

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Geometric Sequences and Series 2 on the next, 1 grain was to be placed on the 1 st square of the chess board, 4 on the 3 rd and so on, doubling the number each square. How many must be placed on the 64 th square?

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Geometric Sequences and Series We have a sequence: Each term is twice the previous term, so by the 64 th term we have multiplied by 2 sixty-three times We have approximately or 9 followed by 18 zeros! Geometric Sequence

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Geometric Sequences and Series The sequence is an example of a Geometric sequence A sequence is geometric if where r is a constant called the common ratio In the above sequence, r = 2

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Geometric Sequences and Series A geometric sequence or geometric progression (G.P.) is of the form The n th term of an G.P. is

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Geometric Sequences and Series Exercises 1. Use the formula for the n th term to find the term indicated of the following geometric sequences (b) (c) (a) Ans:

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Geometric Sequences and Series e.g.1 Evaluate Writing out the terms helps us to recognize the G.P. Summing terms of a G.P. Although with a calculator we can see that the sum is 186, we need a formula that can be used for any G.P. The formula will be proved next but you don’t need to learn the proof.

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Geometric Sequences and Series Subtracting the expressions gives With 5 terms of the general G.P., we have Multiply by r: Move the lower row 1 place to the right Summing terms of a G.P.

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Geometric Sequences and Series Subtracting the expressions gives With 5 terms of the general G.P., we have Multiply by r: and subtract Summing terms of a G.P.

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Geometric Sequences and Series Subtracting the expressions gives With 5 terms of the general G.P., we have Multiply by r: Summing terms of a G.P.

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Geometric Sequences and Series Similarly, for n terms we get So, Take out the common factors and divide by ( 1 – r ) Summing terms of a G.P.

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Geometric Sequences and Series gives a negative denominator if r > 1 The formula Instead, we can use Summing terms of a G.P. To get this version of the formula, we’ve multiplied the 1 st form by

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Geometric Sequences and Series For our series Using Summing terms of a G.P.

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Geometric Sequences and Series Find the sum of the first 20 terms of the geometric series, leaving your answer in index form e.g. 2 Solution: We’ll simplify this answer without using a calculator Summing terms of a G.P.

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Geometric Sequences and Series There are 20 minus signs here and 1 more outside the bracket! Summing terms of a G.P.

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Geometric Sequences and Series e.g. 3 In a geometric sequence, the sum of the 3rd and 4th terms is 4 times the sum of the 1st and 2nd terms. Given that the common ratio is not –1, find its possible values. Solution: As there are so few terms, we don’t need the formula for a sum 3 rd term + 4 th term = 4 ( 1 st term + 2 nd term ) Divide by a since the 1 st term, a, cannot be zero: Summing terms of a G.P.

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Geometric Sequences and Series Using the factor theorem: We need to solve the cubic equation Summing terms of a G.P. Using long division the quadratic factor is r 2 – 4 Since we were told we get Factorizing

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Geometric Sequences and Series Using the factor theorem: We need to solve the cubic equation Since we were told we get Factorizing Summing terms of a G.P.

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Geometric Sequences and Series e.g. 4 £ 100 is invested every year on the first of January and earns compound interest at the rate of 4 % per annum. Find the amount by the end of the 5 th year, to the nearest penny. Solution: The last £ 100 is invested for 1 year only. The 4 th £ 100 is invested for 2 years so at the end is worth At the end, this £ 100 is worth 100 is a common factor Summing terms of a G.P.

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Geometric Sequences and Series (nearest penny) At the end of the 5 years, the total invested will be worth This is a G.P. with

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Geometric Sequences and Series SUMMARY A geometric sequence or geometric progression (G.P.) is of the form The n th term of an G.P. is The sum of n terms is or

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Geometric Sequences and Series Exercises 1. Find the sum of the first 15 terms of the following G.P., giving the answers in index form 2 + 8 + 32 +... 2. Find the sum of the first 15 terms of the G.P. 4 2 + 1 +... giving your answer correct to 3 significant figures.

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Geometric Sequences and Series Exercises 1. Solution: 2 + 8 + 32 +... 2. Solution: 4 2 + 1 +... ( 3 s.f. )

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Geometric Sequences and Series

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The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

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Geometric Sequences and series SUMMARY A geometric sequence or geometric progression (G.P.) is of the form The n th term of an G.P. is The sum of n terms is or

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Geometric Sequences and series Find the sum of the first 20 terms of the geometric series, leaving your answer in index form e.g. Solution: We’ll simplify this answer without using a calculator Summing terms of a G.P.

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Geometric Sequences and series There are 20 minus signs here and 1 more outside the bracket!

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Geometric Sequences and series e.g. In a geometric sequence, the sum of the 3rd and 4th terms is 4 times the sum of the 1st and 2nd terms. Given that the common ratio is not –1, find its possible values. Solution: As there are so few terms, we don’t need the formula for a sum 3 rd term + 4 th term = 4 ( 1 st term + 2 nd term ) Divide by a since the 1 st term, a, cannot be zero: Summing terms of a G.P.

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Geometric Sequences and series Using the factor theorem: We need to solve the cubic equation Since we were told we get Factorizing

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Geometric Sequences and series e.g. £ 100 is invested every year on the first of January and earns compound interest at the rate of 4 % per annum. Find the amount by the end of the 5 th year, to the nearest penny. Solution: The last £ 100 is invested for 1 year only. The 4 th £ 100 is invested for 2 years so at the end is worth At the end, this £ 100 is worth 100 is a common factor

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Geometric Sequences and series (nearest penny) At the end of the 5 years, the total invested will be worth This is a G.P. with

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