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

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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.

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

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

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

A geometric sequence or geometric progression (G.P.) is of the form The nth term of an G.P. is

Exercises 1. Use the formula for the nth term to find the term indicated of the following geometric sequences (a) Ans: (b) Ans: (c) Ans:

The formula will be proved next but you don’t need to learn the proof. Summing terms of a G.P. e.g.1 Evaluate Writing out the terms helps us to recognize the 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.

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

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

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

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

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

Summing terms of a G.P. For our series Using

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

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

3rd term + 4th term = 4( 1st term + 2nd term ) Summing terms of a G.P. 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 3rd term + 4th term = 4( 1st term + 2nd term ) Divide by a since the 1st term, a, cannot be zero:

Summing terms of a G.P. We need to solve the cubic equation Using the factor theorem: Using long division the quadratic factor is r2 – 4 Factorizing Since we were told we get

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

Summing terms of a G.P. 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 5th year, to the nearest penny. Solution: The last £100 is invested for 1 year only. At the end, this £100 is worth 100 is a common factor The 4th £100 is invested for 2 years so at the end is worth

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

SUMMARY A geometric sequence or geometric progression (G.P.) is of the form The nth term of an G.P. is The sum of n terms is or

2. Find the sum of the first 15 terms of the G.P. 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.

2 + 8 + 32 + . . . 4 - 2 + 1 + . . . Exercises 1. Solution: 2 + 8 + 32 + . . . 2. Solution: 4 - 2 + 1 + . . . ( 3 s.f. )

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.

SUMMARY A geometric sequence or geometric progression (G.P.) is of the form The nth term of an G.P. is The sum of n terms is or

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

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

3rd term + 4th term = 4( 1st term + 2nd term ) 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 3rd term + 4th term = 4( 1st term + 2nd term ) Divide by a since the 1st term, a, cannot be zero: Summing terms of a G.P.

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

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 5th year, to the nearest penny. Solution: The last £100 is invested for 1 year only. The 4th £100 is invested for 2 years so at the end is worth At the end, this £100 is worth 100 is a common factor

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