GEOMETRIC SERIES
1. Geometric Progressions (G.P.s)
In a geometric progression each term is obtained by multiplying the previous term by a constant, known as the common ratio.
• 3, 6, 12, 24, 48, ... is a G.P. with first term 3 and common ratio 2.
• 4, –12, 36, –108, 324, ... is a G.P. with first term 4 and common ratio –3.
A G.P. with first term a and common ratio r has terms
Example1:The second term of a G.P. is 2, and the fifth term is 54. Find the twelfth term.
Dividing,
Example2:The first three terms of a G.P are , and . Find the two possible values of x and the corresponding common ratios.
We compare two expressions for the common ratio, which is the ratio of consecutive terms.
If , we have the G.P. 5, 10, 20... which has common ratio 2.
If , we have the G.P. –22.5, 37.5, –62.5... which has common ratio .
Example3:I invest £300 in the bank at 4% interest per year.
a) How much will I have after six years?
b) How many years must pass before I have £1000?
a) The amount of money is multiplied by 1.04 each year. After six years I will have
b) Using trial and improvement,
So it will take 31 years.
A rather better method is to use logarithms.
Example 4:The population of Bengal tigers is falling by 7% per year. How long will it take for the population to halve in size?
The number of tigers is multiplied by a common ratio of 0.93 each year.
So it will take ten years. Notice the inequality sign reverses when we divide by log 0.93, as this is a negative number
Example 5 : At which term does the sequence 2, 3, 4.5, 6.75... first exceed 2000?
This is a G.P. with first term 2 and common ratio 1.5. The nth term is therefore .
We require
So the least possible value of n is 19. The 19th term is
Example6:The first term of an arithmetic series is 2 and the terms are not all equal. The first, third and eleventh terms of this series are also the first, second and third terms respectively of a geometric series. Find the common difference of the arithmetic series.
We have
This generates two simultaneous equations.
Comparing expressions for ,
We reject the solution since the terms are not all equal. Therefore and .
Check :
A.P. 2 5 8 11 14 17 20 23 26 29 32
G.P. 2 8 32
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2. The Sum of a Geometric Progression
Suppose we wish to find the sum of the first eight terms of a geometric progression, with first term 5 and common ratio 3.
Multiplying by 3,
Notice that the majority of the terms ‘pair up’. Subtracting the first series from the second,
We can use this method to obtain a formula for the first n terms of a G.P. with first term a and common ratio r.
Therefore we have
The second form is simpler when r < 1, as both numerator and denominator will be positive.
Example 1 : Find the sum of the first eleven terms of the series 5 + 10 + 20 + 40 + ....
We have a = 5, r = 2, n = 11.
Example 2 : Find the sum of the first thirty terms of the series 3 – 4.5 + 6.75 – 10.125 ...
We have a = 3, r = –1.5, n = 30.
N.B. Be careful to type (–1.5) ^ 30 rather than –1.5 ^ 30 into your calculator here. BIDMAS!
Example 3 : Evaluate
This as a G.P. with , and .
Example4:A 100cm length of string is cut into five pieces whose lengths form a geometric progression. The fourth piece is 4 times as long as the second piece. Find a formula for the length of the nth piece, and hence the length of the longest piece.
We reject the solution , as the length of a piece of string cannot be negative!
Now
Example 5 : Find the least value of n such that the sum 4 + 12 + 36 + 108 + … to n terms would first exceed a million.
We require
So
Example6:A devious and mathematically astute teenager puts the following proposition to his dim Dad. Instead of the usual £5 pocket money a week, he suggests he be given 1p the first day, 2p the second, 4p the third, 8p the fourth and so on for the next four weeks. His Dad agrees. How much does he end up shelling out? (It is illuminating to get the class to guess first!)
We have a G.P. with a = 1, r = 2, n = 28.
Example7:An investor deposits £300 into his bank on January 1st each year. The interest paid is 7% per annum, credited to the account at the end of the year. How much is in the account after 20 years? How much interest has been paid altogether?
This is a common A-level type question, and is best looked at by considering the ‘life story’ of each £300.
The first £300 has attracted interest twenty times and is therefore worth .
The second £300 has attracted interest nineteen times and is therefore worth .
This continues until the last £300 which has attracted interest once and is therefore worth .
Reversing these terms, we have a G.P. with , , .
The amount of interest paid is .
Example8:Each year a farmer plants 200 saplings in a field. She estimates that each year 15% of her plants will die. How many live saplings will she have after fifteen years? How many will have died altogether?
Again, we consider the life story of each batch of 200 saplings.
Of the first 200 saplings, only have survived.
Of the second batch, have survived, and so on until the last batch, of which have survived.
So our G.P. is .
Reversing these terms, we have a G.P. with , , .
The number of saplings that have died is .
Finally in this section, note that the formula can make things unnecessarily complicated when you are only summing two or three terms…
Example 9 : The sum of the first three terms of a G.P. is 21. The first term is 12. Find the possible values of the common ratio.
We could do this…
…but now we have a cubic equation to solve. Far better is to realise that the first three terms are 12, , and , and so we have…
Checking, with we have ; and with we have
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3. Convergent Series
Activity2:Do the Frog Olympics worksheet. Important : work out all distances in fractions. Decimals get messy and time-consuming very quickly.
Consider the G.P. with first term 1 and common ratio .
We can see that, despite being infinite, this series adds up to a finite sum, namely 2, because each term fills up half the remaining ‘gap’. Because the sum has a finite value, we say the series converges.
Clearly, if a series is to converge, its terms must become smaller and smaller. Therefore a G.P. with common ratio greater than 1 or less than –1 will not converge, and
• No A.P. will converge.
• No G.P. with a common ratio greater than or equal to 1 will converge.
• No G.P. with a common ratio less than or equal to –1 will converge. In fact, a G.P. with a common ratio –1 has a sum that oscillates as more terms are taken. For example the G.P. 3, –3, 3, –3, 3 ... has a sum which oscillates between 3 and 0.
4. The Sum to Infinity of a Geometric Progression
We have already seen that the sum of the first n terms of a geometric progression is given by
Now if , will become smaller and smaller as n increases, which we write as . Otherwise . Therefore
Example 1 : Which of the following series converge? For those that do, sum them to infinity.
a) 1 + 6 + 11 + 16 + 21 + ...
b)
c)
d) 3 + 6 + 12 + 24 + ...
a) An A.P. which does not converge.
b) A G.P. with and .
c) A G.P. with and .
d) A G.P. with r = 2, so the series does not converge.
Example 2 : Evaluate
This as an infinite, convergent G.P. with and .
Example3:A rubber ball is dropped from a height of 0·5m. The height of each bounce is 70% of the height of the previous bounce. How far does the ball travel in total?
The rises and falls come in pairs, apart from the initial fall which has no corresponding rise. So we double the G.P. with , and , and then subtract 0.5m.
Example4:Janet and John are playing a dice game. They take it turns to roll a dice. The first person to roll a 6 wins the game. If Janet throws the dice first, what is the probability that she will win?
Let S represent the event ‘the score is 6’ and N the event ‘the score is not 6’. For Janet to win, the first 6 has to be on the first, third, fifth etc. rolls. Therefore
This is a G.P. with , and .
Example5:The sum of the first five terms of a G.P. is 46.5. The sum to infinity of the G.P. is 48. Find the first term and the common ratio of the series.
We can write down two equations from the information in the question.
Substituting for ,
Substituting into the second equation,
C2 p105 Ex 7E Topic Review : Geometric Series