Population, when unchecked, increases in a geometrical ratio. Subsistence increases only in an arithmetical ratio. a slight acquaintance with numbers will show the immensity of the first power in comparison with the second.
Each of the following sequences is related to one of the pictures above.
(i) 5000, 10 000, 20 000, 40 000, … .
(ii) 8, 0, 10, 10, 10, 10, 12, 8, 0, … .
(iii) 5, 3.5, 0, –3.5, –5, –3.5, 0, 3.5, 5, 3.5, … .
(iv) 20, 40, 60, 80, 100, … .
(a) Identify which sequence goes with which picture.
(b) Give the next few numbers in each sequence.
(c) Describe the pattern of the numbers in each case.
(d) Decide whether the sequence will go on for ever, or come to a stop.
Definitions and notation
A sequence is a set of numbers in a given order, like
Each of these numbers is called a term of the sequence. When writing the terms of a sequence algebraically, it is usual to denote the position of any term in the sequence by a subscript, so that a general sequence might be written:
For the sequence above, the first term is and so on.
When the terms of a sequence are added together, like
the resulting sum is called a series. The process of adding the terms together is called summation and indicated by the symbol (the Greek letter sigma), with the position of the first and last terms involved given as limits.
In cases like this one, where there is no possibility of confusion, the sum would normally be written more simply as
If all the terms were to be summed, it would usually be denoted even more simply,
A sequence may have an infinite number of terms, in which case it is called an infinite sequence. The corresponding series is called an infinite series.
In mathematics, although the word series can describe the sum of the terms of any sequence, it is usually used only when summing the sequence provides some useful or interesting overall result.
The phrase ‘sum of a sequence’ is often used to mean the sum of the terms of a sequence (i.e. the series).
Any ordered set of numbers, like the scores of this golfer on an 18-hole round (see figure 3.1) form a sequence. In mathematics, we are particularly interested in those which have a well-defined pattern, often in the form of an algebraic formula linking the terms. The sequences you met at the start of this chapter show various types of pattern.
A sequence in which the terms increase by the addition of a fixed amount (or decrease by the subtraction of a fixed amount), is described as arithmetic. The increase from one term to the next is called the common difference.
common difference 3.
This sequence can be written algebraically as
As successive terms of an arithmetic sequence increase (or decrease) by a fixed amount called the common difference, d, you can define each term in the sequence in relation to the previous term:
When the terms of an arithmetic sequence are added together, the sum is called an arithmetic progression, often abbreviated to A.P. An alternative name is an arithmetic series.
When describing arithmetic progressions and sequences in this book, the following conventions will be used:
Thus in the arithmetic sequence 5, 7, 9, 11, 13, 15, 17,
a = 5, l = 17, d = 2 and n = 7.
The terms are formed as follows.
You can see that any term is given by the first term plus a number of differences.
The number of differences is, in each case, one less than the number of the term. You can express this mathematically as
For the last term, this becomes
l = a + (n − 1)d.
These are both general formulae which apply to any arithmetic sequence.
Find the 17th term in the arithmetic sequence 12, 9, 6, … .
In this case a = 12 and d = −3.
How many terms are there in the sequence 11, 15, 19, …, 643?
This is an arithmetic sequence with first term a = 11, last term l = 643 and common difference d = 4.
The relationship l = a + (n − 1)d may be rearranged to give
This gives the number of terms in an A.P. directly if you know the first term, the last term and the common difference.
The sum of the terms of an arithmetic progression
When Carl Friederich Gauss (1777−1855) was at school he was always quick to answer mathematics questions. One day his teacher, hoping for half an hour of peace and quiet, told his class to add up all the whole numbers from 1 to 100.
Almost at once the 10-year-old Gauss announced that he had done it and that the answer was 5050.
Gauss had not of course added the terms one by one. Instead he wrote the series down twice, once in the given order and once backwards, and added the two together:
The numbers 1, 2, 3, … , 100 form an arithmetic sequence with common difference
Gauss’ method can be used for finding the sum of any arithmetic series.
It is common to use the letter S to denote the sum of a series. When there is any doubt as to the number of terms that are being summed, this is indicated by a
Find the value of 8 + 6 + 4 + … + (−32).
This is an arithmetic progression, with common difference −2. The number of terms, n, may be calculated using
The sum S of the progression is then found as follows.
Since there are 21 terms, this gives 2S = −24 × 21, so S = −12 × 21 = −252.
Generalising this method by writing the series in the conventional notation gives:
Since there are n terms, it follows that
This result may also be written as
Find the sum of the first 100 terms of the progression
In this arithmetic progression
Jamila starts a part-time job on a salary of $9000 per year, and this increases by an annual increment of $1000. Assuming that, apart from the increment, Jamila’s salary does not increase, find
(i) her salary in the 12th year
(ii) the length of time she has been working when her total earnings are $100 000.
Jamila’s annual salaries (in dollars) form the arithmetic sequence
9000, 10 000, 11 000, … .
with first term a = 9000, and common difference d = 1000.
i) Her salary in the 12th year is calculated using:
(ii) The number of years that have elapsed when her total earnings are $100 000 is given by:
The root n = −25 is irrelevant, so the answer is n = 8.
Jamila has earned a total of $100 000 after eight years.
A human being begins life as one cell, which divides into two, then four… .
The terms of a geometric sequence are formed by multiplying one term by a fixed number, the common ratio, to obtain the next. This can be written inductively as:
The sum of the terms of a geometric sequence is called a geometric progression, shortened to G.P. An alternative name is a geometric series.
When describing geometric sequences in this book, the following conventions are used:
Thus in the geometric sequence 3, 6, 12, 24, 48,
a = 3, r = 2 and n = 5.
The terms of this sequence are formed as follows.
You will see that in each case the power of r is one less than the number of the term: and 4 is one less than 5. This can be written deductively as
These are both general formulae which apply to any geometric sequence.
Given two consecutive terms of a geometric sequence, you can always find the common ratio by dividing the later term by the earlier. For example, the geometric sequence … 5, 8, … has common ratio
Find the seventh term in the geometric sequence 8, 24, 72, 216, … .
In the sequence, the first term a = 8 and the common ratio r = 3.
The kth term of a geometric sequence is given by
How many terms are there in the geometric sequence 4, 12, 36, … , 708 588?
Since it is a geometric sequence and the first two terms are 4 and 12, you can immediately write down
The third term allows you to check you are right.
You can use logarithms to solve an equation like this, but since you know that n is a whole number it is just as easy to work out the powers of 3 until you come to 177 147.
So n – 1 = 11 and n = 12.
There are 12 terms in the sequence.
How would you use a spreadsheet to solve the equation
The sum of the terms of a geometric progression
The origins of chess are obscure, with several countries claiming the credit for its invention. One story is that it came from China. It is said that its inventor presented the game to the Emperor, who was so impressed that he asked the
inventor what he would like as a reward.
‘One grain of rice for the first square on the board, two for the second, four for the third, eight for the fourth, and so on up to the last square’, came the answer.
The Emperor agreed, but it soon became clear that there was not enough rice in the whole of China to give the inventor his reward.
How many grains of rice was the inventor actually asking for?
The answer is the geometric series with 64 terms and common ratio 2:
This can be summed as follows.
Call the series S:
Now multiply it by the common ratio, 2:
The total number of rice grains requested was therefore (which is about
How many tonnes of rice is this, and how many tonnes would you expect there to be in China at any time?
(One hundred grains of rice weigh about 2 grammes. The world annual production of all cereals is about
The method shown above can be used to sum any geometric progression.
Example 3.8 Find the value of 0.2 + 1 + 5 + … + 390 625.
This is a geometric progression with common ratio 5.
The same method can be applied to the general geometric progression to give a formula for its value:
Infinite geometric progressions
Clearly the more terms you take, the nearer the sum gets to 2. In the limit, as the number of terms tends to infinity, the sum tends to 2.
This is an example of a convergent series. The sum to infinity is a finite number.
You can see this by substituting n the formula for the sum of the series:
The larger the number of terms, n, the smaller becomes and so the nearer is to the limiting value of 2 (see figure 3.3). Notice that can never be negative, however large n becomes; so can never exceed 2.
In the general geometric series the terms become progressively
smaller in size if the common ratio r is between −1 and 1. This was the caseabove: r had the value 1
In such cases, the geometric series is convergent.
If, on the other hand, the value of r is greater than 1 (or less than −1) the terms in the series become larger and larger in size and so the series is described as divergent.
A series corresponding to a value of r of exactly +1 consists of the first term a repeated over and over again. A sequence corresponding to a value of r of exactly −1 oscillates between +a and −a. Neither of these is convergent.
It only makes sense to talk about the sum of an infinite series if it is convergent. Otherwise the sum is undefined.
The condition for a geometric series to converge, −1 < r < 1, ensures that as and so the formula for the sum of a geometric series:
may be rewritten for an infinite series as:
Find the sum of the terms of the infinite progression 0.2, 0.02, 0.002, … .
This is a geometric progression with a = 0.2 and r = 0.1.
Its sum is given by
You may have noticed that the sum of the series 0.2 + 0.02 + 0.002 + … is 0. ˙2, and that this recurring decimal is indeed the same as
The first three terms of an infinite geometric progression are 16, 12 and 9.
(i) Write down the common ratio.
(ii) Find the sum of the terms of the progression.
(i) The common ratio is
(ii) The sum of the terms of an infinite geometric progression is given by:
Draw an equilateral triangle with sides 9 cm long.
Trisect each side and construct equilateral triangles on the middle section of each
side as shown in diagram (b).
Repeat the procedure for each of the small triangles as shown in (c) and (d) so that
you have the first four stages in an infinite sequence.
Calculate the length of the perimeter of the figure for each of the first six steps,
starting with the original equilateral triangle.
What happens to the length of the perimeter as the number of steps increases?
Does the area of the figure increase without limit?
Achilles and the tortoise
Achilles (it is said) once had a race with a tortoise. The tortoise started 100 m
ahead of Achilles and moved at compared to Achilles’ speed of 10
Achilles ran to where the tortoise started only to see that it had moved 1 m further on. So he ran on to that spot but again the tortoise had moved further on, this time by 0.01 m. This happened again and again: whenever Achilles got to the spot where the tortoise was, it had moved on. Did Achilles ever manage to catch the tortoise?
A special type of series is produced when a binomial (i.e. two-part) expression like (x + 1) is raised to a power. The resulting expression is often called a binomial expansion.
The simplest binomial expansion is (x + 1) itself. This and other powers of (x + 1) are given below.
If you look at the coefficients on the right-hand side above you will see that they form a pattern.
This is called Pascal’s triangle, or the Chinese triangle. Each number is obtained by adding the two above it, for example
This pattern of coefficients is very useful. It enables you to write down the expansions of other binomial expressions. For example,
Write out the binomial expansion of
The binomial coefficients for power 4 are 1 4 6 4 1.
In each term, the sum of the powers of x and 2 must equal 4.
So the expansion is
Write out the binomial expansion of
The binomial coefficients for power 5 are 1 5 10 10 5 1.
The expression (2a − 3b) is treated as (2a + (−3b)).
So the expansion is
Blaise Pascal has been described as the greatest might-have-been in the history of mathematics. Born in France in 1623, he was making discoveries in geometry by the age of 16 and had developed the first computing machine before he was 20.
Pascal suffered from poor health and religious anxiety, so that for periods of his life he gave up mathematics in favour of religious contemplation. The second of these periods was brought on when he was riding in his carriage: his runaway horses dashed over the parapet of a bridge, and he was only saved by the miraculous breaking of the traces. He took this to be a sign of God’s disapproval of his mathematical work. A few years later a toothache subsided when he was thinking about geometry and this, he decided, was God’s way of telling him to return to
Pascal’s triangle (and the binomial theorem) had actually been discovered by Chinese mathematicians several centuries earlier, and can be found in the works of Yang Hui (around 1270 a.d.) and Chu Shi-kie (in 1303 a.d.). Pascal is remembered for his application of the triangle to elementary probability, and for his study of the relationships between binomial coefficients.
Pascal died at the early age of 39.
Tables of binomial coefficients
Values of binomial coefficients can be found in books of tables. It is helpful to use these when the power becomes large, since writing out Pascal’s triangle becomes progressively longer and more tedious, row by row.
Write out the full expansion of
The binomial coefficients for the power 10 can be found from tables to be
1 10 45 120 210 252 210 120 45 10 1
and so the expansion is
As the numbers are symmetrical about the middle number, tables do not always give the complete row of numbers.
The formula for a binomial coefficient
There will be times when you need to find binomial coefficients that are outside the range of your tables. The tables may, for example, list the binomial coefficients for powers up to 20. What happens if you need to find the coefficient
Clearly you need a formula that gives binomial coefficients.
The first thing you need is a notation for identifying binomial coefficients. It is usual to denote the power of the binomial expression by n, and the position in the row of binomial coefficients by r, where r can take any value from 0 to n. So for row 5 of Pascal’s triangle
The general binomial coefficient corresponding to values of n and r is written as An alternative notation is which is said as
The next step is to find a formula for the general binomial coefficient
However, to do this you must be familiar with the term factorial.
The quantity ‘8 factorial’, written 8!, is
8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40 320.
Similarly, 12! = 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 479 001 600,
and n! = n × (n − 1) × (n − 2) × … × 1, where n is a positive integer.
The table shows an alternative way of laying out Pascal’s triangle.
Show that by following the procedure below.
The numbers in column 0 are all 1.
To find each number in column 1 you multiply the 1 in column 0 by the row
(i) Find, in terms of n, what you must multiply each number in column 1 by to find the corresponding number in column 2.
(ii) Repeat the process to find the relationship between each number in column 2 and the corresponding one in column 3.
(iii) Show that repeating the process leads to
(iv) Show that this can also be written as
You can see that these numbers, 1, 5, 10, 10, 5, 1, are row 5 of Pascal’s triangle.
Most scientific calculators have factorial buttons, e.g. Many also have
buttons. Find out how best to use your calculator to find binomial coefficients , as well as practising non-calculator methods.
Find the coefficient of in the expansion of
Notice how 17! was cancelled in working out Factorials become large numbers
very quickly and you should keep a look-out for such opportunities to simplify
The expansion of
When deriving the result for you found the binomial coefficients in the form.
This form is commonly used in the expansion of expressions of the type
Use the binomial expansion to write down the first four terms, in ascending powers of
The expression is said to be in ascending powers of x, because the powers of x are increasing from one term to the next.
An expression like is in descending powers of x, because the powers of x are decreasing from one term to the next.
Use the binomial expansion to write down the first four terms, in ascending powers of Simplify the terms.
Think of Keep the brackets while you write out the terms.
The first three terms in the expansion of
powers of x, are Find the values of a, b and c.
Find the first three terms in the expansion in terms of a and b:
What is the connection between your results and the coefficients in Pascal’s triangle?
Relationships between binomial coefficients
There are several useful relationships between binomial coefficients.
Because Pascal’s triangle is symmetrical about its middle, it follows that
You have seen that each term in Pascal’s triangle is formed by adding the two
above it. This is written formally as
Thus the sum of the binomial coefficients for power n is
The binomial theorem and its applications
The binomial expansions covered in the last few pages can be stated formally as
the binomial theorem for positive integer powers:
Notice the use of the summation symbol The right-hand side of the statement
reads ‘the sum of for values of r from 0 to n’.
It therefore means
The binomial theorem is used on other types of expansion and it has applications in many areas of mathematics.
The binomial distribution
In some situations involving repetitions of trials with two possible outcomes, the probabilities of the various possible results are given by the terms of a binomial expansion. This is covered in Probability and Statistics 1.
The number of ways of selecting r objects from n (all different) is given by
This is also covered in Probability and Statistics 1.
Routes to victory
In a recent soccer match, Juventus beat Manchester United 2–1.
What could the half-time score have been?
(i) How many different possible half-time scores are there if the final score is
2–1? How many if the final score is 4–3?
(ii) How many different ‘routes’ are there to any final score? For example, for the
above match, putting Juventus’ score first, the sequence could be:
0–0 → 0–1 → 1–1 → 2–1
or 0–0 → 1–0 → 1–1 → 2–1
or 0–0 → 1–0 → 2–0 → 2–1.
So in this case there are three routes.
Investigate the number of routes that exist to any final score (up to a maximum
of five goals for either team).
Draw up a table of your results. Is there a pattern?