Geometric Sequences and Series

Introduction In Core 1 you learn about Arithmetic Sequences, where the pattern is based around an addition/subtraction rule For example, a sequence with a common difference of 2 (5, 7, 9, 11…) is an Arithmetic Sequence This Chapter focuses on sequences where the pattern is based around multiplication or division A sequence such as 2, 4, 8, 16, 32 is known as a Geometric Sequence

Teachings for Exercise 7A

Geometric Sequences and Series You need to be able to spot patterns to work out the rule for a Geometric Sequence In any Geometric sequence, u1, u2, u3 … un Is the common ratio, and is the same for all consecutive pairs Example Questions Calculate the common ratio for each of the following sequences… a) 3, 12, 48, 192 b) 54, 18, 6, 2 etc… 7A

Geometric Sequences and Series You need to be able to spot patterns to work out the rule for a Geometric Sequence In any Geometric sequence, u1, u2, u3 … un Is the common ratio, and is the same for all consecutive pairs Example Questions Calculate the common ratio for each of the following sequences… c) 5, x, 51.2 Multiply by 5 Multiply by x Square root etc… 7A

Teachings for Exercise 7B

Geometric Sequences and Series You can define a Geometric Sequence using a first term ‘a’ and a common ratio ‘r’ a, ar, ar2, ar3, …, arn-1 Example Questions Find the nth and 10th terms of the following sequences… a) 3, 6, 12, 24… First Term = 3 1st Term 2nd Term 3rd Term 4th Term nth Term Common Ratio = 2 Nth Term = 3 x 2n-1 10th Term = 3 x 29 = 1536 7B

Geometric Sequences and Series You can define a Geometric Sequence using a first term ‘a’ and a common ratio ‘r’ a, ar, ar2, ar3, …, arn-1 Example Questions Find the nth and 10th terms of the following sequences… b) 40, -20, 10, -5… First Term = 40 1st Term 2nd Term 3rd Term 4th Term nth Term Common Ratio = -0.5 Nth Term = 40 x (-0.5)n-1 10th Term = 40 x (-0.5)9 = -5/64 7B

Geometric Sequences and Series You can define a Geometric Sequence using a first term ‘a’ and a common ratio ‘r’ a, ar, ar2, ar3, …, arn-1 Example Questions The second term of a Geometric sequence is 4, and the 4th term is 8. Find the values of the common ratio and the first term 1 2nd Term  2 4th Term  1st Term 2nd Term 3rd Term 4th Term nth Term 2 ÷ 1  Square root Sub r into 1 Divide by √2 Rationalise 7B

Geometric Sequences and Series You can define a Geometric Sequence using a first term ‘a’ and a common ratio ‘r’ a, ar, ar2, ar3, …, arn-1 Example Questions The numbers 3, x, and (x + 6) form the first three terms of a positive geometric sequence. Calculate the 15th term of the sequence 1st Term 2nd Term 3rd Term 4th Term nth Term Cross Multiply by 3 and x Set equal to 0 Factorise x has to be positive 7B

Geometric Sequences and Series You can define a Geometric Sequence using a first term ‘a’ and a common ratio ‘r’ a, ar, ar2, ar3, …, arn-1 Example Questions The numbers 3, x, and (x + 6) form the first three terms of a positive geometric sequence. Calculate the 15th term of the sequence 1st Term 2nd Term 3rd Term 4th Term nth Term First term = 3 Common Ratio = 2 Nth term = 3 x 2n-1 15th Term = 3 x 214 15th Term = 49152 7B

Teachings for Exercise 7C

Geometric Sequences and Series You need to be able to use Percentages in Geometric Sequences  If I was to increase an amount by 10%, what would I multiply the value by?  1.1  If I was to increase an amount by 17%, what would I multiply by?  1.17 Example Question £A is to be invested in a savings fund at a rate of 4%. How much should be invested so the fund is worth £10,000 in 5 years? Y1 Y2 Y3 Y4 Y5 A Ar Ar2 Ar3 Ar4 Ar5 A = A r = 1.04 Ar5 = 10,000 r = 1.04 A x 1.045 = 10,000 ÷ 1.045 A = 10,000 1.045 A = £8219.27 7C

Geometric Sequences and Series You will need to be able to apply logarithms to solve problems  Remember that logarithms are used when solving equations where the power is unknown.  You will need to remember ‘the power law’ in most questions of this type Example Question What is the first term in the sequence 3, 6, 12, 24… to exceed 1 million? Sub in ‘a’ and ‘r’ Divide by 3 Take logs Work out the right side Use the power law Divide by log(2) Add 1 It must be the 20th term 7C

Teachings for Exercise 7D

Geometric Sequences and Series You need to be able to work out the sum of a Geometric Sequence 1 Multiply all terms by r 2 1 - 2 Factorise both sides Divide by (1 - r) 7D

Geometric Sequences and Series You need to be able to work out the sum of a Geometric Sequence Example Question Find the sum of the following series: 1024 – 512 + 256 – 128 + … + … + 1 Substitute Divide by 1024 ??? Divide by log(0.5) Take logs Work out the LHS Use the power law Add 1 7D

Geometric Sequences and Series You need to be able to work out the sum of a Geometric Sequence Example Question Find the sum of the following series: 1024 – 512 + 256 – 128 + … + … + 1 Substitute Work it out! 7D

Geometric Sequences and Series You need to be able to work out the sum of a Geometric Sequence Example Question Find the value of n at which the sum of the following sequence is greater than 2,000,000 1 + 2 + 4 + 8… + … + The sum must be above 2,000,000 Substitute Take logs Multiply by -1 (and reverse the sign) The power law Divide by log(2) Subtract 1 Work out the RHS Multiply by -1 (and reverse the sign) 7D

Geometric Sequences and Series You need to be able to work out the sum of a Geometric Sequence Example Question Find the value of the following: The value or r to put in for the last number The formula of the sequence… The value or r to put in for the first number Work out the first, second and last terms… Substitute Work it out! 7D

Teachings for Exercise 7E

Geometric Sequences and Series You need to be able to work out the sum to infinity of a Geometric Sequence Consider the sequence with the following formula: This sequence CONVERGES to 0.1 recurring… First 4 terms As Decimals A Sequence will converge if the common ratio, r is between -1 and 1. Sum of 1st term Sum of 1st and 2nd terms Sum of 1st to 3rd terms Sum of 1st to 4th terms 7E

Geometric Sequences and Series You need to be able to work out the sum to infinity of a Geometric Sequence Example Question Find the sum to infinity of the following sequence: 40 + 10 + 2.5 + 0.625… Substitute Work it out! 7E

Geometric Sequences and Series You need to be able to work out the sum to infinity of a Geometric Sequence Example Question The Sum to infinity of a Sequence is 16, and the sum of the first 4 terms is 15. Find the possible values of r, and the first term if all terms are positive… Sub in 15, and n = 4 Replace a Cancel out (1 - r) Sub in 16 Divide by 16 Rearrange in terms of a Subtract 1 7E

Summary We have learnt what a Geometric Sequence is We have seen how to calculate the common ratio and first term We have also looked at the sum of a Geometric Sequence, as well as the Sum to Infinity