Oxford MAT Livestream 2022 – Graphs & Transformations
MAT syllabus
The graphs of quadratics and cubics. Graphs of
sin x,
cos x,
p
x,
ax ,
loga x.
y = f (x
a),
y = f (x) + a
tan x,
Solving equations and inequalities with graphs.
The relations between the graphs
y = f (ax),
y = af (x),
and the graph of y = f (x).
Revision
• The graph of an equation involving x and y is all the points in the (x, y) plane that
satisfy the equation. For a function f (x), the graph of y = f (x) shows the value of f
at each value of x.
• Quadratics y = ax2 + bx + c have graphs like these
y
y
x
x
• Cubics y = ax3 + bx2 + cx + d can have 0 or 1 or 2 turning points.
y
y
x
y
x
x
• Other polynomials have graphs that might have more turning points (up to (n
turning points if xn is the highest power of x in the polynomial)
1)
• Graphs of y = sin x (red solid line) and y = cos x (green dashed line) and y = tan x
(blue dot-dashed line);
For solutions see www.maths.ox.ac.uk/r/matlive
Oxford MAT Livestream 2022 – Graphs & Transformations
2 y
1
x
90
180
270
360
1
2
p
• Here are some more graphs. Note that x = x1/2 so the derivative is 12 x 1/2 , which
gets arbitrarily large near x = 0.
p
y= x
y = ax with a > 1
y = ax with 0 < a < 1
2 y
1.5
4
y
4
y
1
2
0.5
2
x
x
1
2
2
3
x
2
2
2
• Here’s the graph of loga x. Note that loga x is very negative for x close to zero.
4
y
2
x
1
2
3
4
5
2
4
• The graph of y = f (x a) is the translation of the graph of y = f (x) by a distance a
in the positive x-direction.
• The graph of y = f (x) + a is the translation of the graph of y = f (x) by a distance a
in the positive y-direction.
• The graph of y = f (ax) is a stretch of the graph of y = f (x) by a factor of
to the x-axis.
1
a
parallel
• The graph of y = af (x) is a stretch of the graph of y = f (x) by a factor of a parallel
to the y-axis.
For solutions see www.maths.ox.ac.uk/r/matlive
Oxford MAT Livestream 2022 – Logarithms and powers
MAT syllabus
Laws of logarithms and exponentials. Solution of the equation ax = b.
Revision
• am an = am+n for any positive real number a and any real numbers m and n.
• (am )n = amn for any positive real number a and any real numbers m and n.
• a
n
=
1
for any positive real number a and any real number n.
an
• a0 = 1 for any non-zero real number a.
• The solution to ax = b where a and b are positive numbers (with a 6= 1) is loga (b). In
this expression, the number a is called the base of the logarithm.
• loga (x) is a function of x which is defined when x > 0. Like with sin x, sometimes the
brackets are omitted if it’s clear what the function is being applied to, so we might
write loga x.
• loga x doesn’t repeat any values; if loga x = loga y then x = y.
• Note the special case loga a = 1 because loga a is the solution x to the equation ax = a,
and that solution is 1.
• In fact, loga (ax ) = x.
• In that sense, the logarithm function is the inverse function for y = ax .
• aloga x = x.
• loga (xy) = loga (x) + loga (y).
• loga (xk ) = k loga x including loga
1
=
x
loga x.
• There’s a mathematical constant called e, which is just a number (it’s about 2.7).
• ex is called the exponential function.
• The laws of indices and laws of logarithms above hold when the base a is equal to e.
• loge x is sometimes written as ln x and the function is sometimes called the natural
logarithm.
For solutions see www.maths.ox.ac.uk/r/matlive
Oxford MAT Livestream 2022 – Integration & Di↵erentiation
MAT syllabus
Derivative of xa , including for fractional exponents. Derivative of ekx . Derivative of a sum of
functions. Tangents and normals to graphs. Turning points. Second order derivatives. Maxima and minima. Increasing and decreasing functions. Di↵erentiation from first principles.
Indefinite integration as the reverse of di↵erentiation. Definite integrals and the signed areas
they represent. Integration of xa (where a 6= 1) and sums thereof.
Revision
• The derivative of xa is axa 1 , including for fractional exponents like a = 12 .
• If k is a constant then the derivative of ekx is kekx .
• If a is a constant then the derivative of af (x) is a times the derivative of f (x).
• The derivative of y1 + y2 is (the derivative of y1 )+(the derivative of y2 ). Perhaps this
looks too obvious to need stating, but remember that, for example, the square of y1 +y2
is not equal to (the square of y1 )+(the square of y2 ).
• The tangent to a graph at a particular point is a line which has the same value and
derivative as the graph at that point. So if we want the tangent to the graph y = x2
at x = 3, we need the value of y (which is 9), and the value of the derivative (which is
6). The derivative of a line is its gradient, so we can write y = 6x + c and solve for c
using the value at x = 3 to get y = 6x 9.
• The normal to a graph is a line which has the same value and is at right angles to the
tangent. Two lines are at right angles if their gradients multiply to 1. So at the point
above, we would want y = 16 x + c and, since the line goes through (3, 9), we have
c = 19
.
2
• If the derivative changes sign (+/ ) at a point, that’s a turning point. You’ll have
zero derivative at the turning point, but that’s not actually sufficient for the derivative
to change sign (e.g. x3 has zero derivative at x = 0, but that’s not a turning point
because the derivative is positive on both sides). A point with zero derivative is called
a stationary point.
• The derivative of a derivative is called the second derivative. You can work out the
derivatives one at a time. So the second derivative of xa would be the derivative
of axa 1 , which is a(a 1)xa 2 . The second derivative of ekx is k 2 ekx . The second
derivative is the rate of change of the derivative.
• “Maxima” is the plural of “maximum”. “Minima” is the plural of “minimum”. A
turning point is a local maximum if the second derivative is negative at that point,
For solutions see www.maths.ox.ac.uk/r/matlive
Oxford MAT Livestream 2022 – Integration & Di↵erentiation
or it’s a local minimum if the second derivative is positive. The word ”local” here
means that very near to that point, the function takes its maximum value. Overall,
the function might have several local maxima, or none, and it might increase without
bound (like y = x for example) so just having derivative zero might not mean that
that’s biggest value of the function. Over an interval like [0, 1] the function might take
its maximum at a local maximum, or maybe at one of the endpoints (like how y = x
would take its maximum value at x = 1 over that interval).
• If the derivative is positive, that’s an increasing function. If it’s negative, that’s a
decreasing function. In general a function might increase in some regions and decrease
in other regions.
• If you have two points on a graph, you can join the line between them – that’s called
the chord. If you move the second point closer and closer to the first point, then the
gradient of the chord gets closer and closer to the gradient of the tangent, which is the
value of the derivative at that point. Calculating the gradient of the chord is a nice
and sensible thing to do; it’s just xy22 yx11 , so this is called a “first principles” approach
to di↵erentiation.
R
• Indefinite integration (without limits as in x2 dx) is the reverse of di↵erentiation in
the sense that if the derivative of f (x) is g(x) then the indefinite integral of g(x) is
f (x) + c where c could be any constant. You can use this to integrate any function
which you could have got as the result of some di↵erentiation.
• The integral of xn is
xn+1
,
n+1
provided that n 6= 1.
R2
Rb
• A definite integral (with limits as in 1 x2 dx) is written like a f (x) dx where a and b
are the two end-points. This is the di↵erence in value of the indefinite integral at the
two end-points; F (b) F (a) where the derivative of F (x) is f (x).
Rb
• If f (x) > 0 for a < x < b then a f (x) dx is the area of the region bounded by the
curve y = f (x), the x-axis, and the lines x = a and x = b.
Rb
• If f (x) < 0 for a < x < b then a f (x) dx is minus one times the area of the region
bounded by the curve y = f (x), the x-axis, and the lines x = a and x = b. Areas are
supposed to be positive. The integral here is sometimes called the “signed area” to
reflect the fact that it’s got a minus sign.
• If f (x) is sometimes positive and sometimes negative in a < x < b then we can split
into separate regions where f (x) is positive or negative before applying the above.
Rb
Ra
• a f (x) dx =
f (x) dx
b
R1
• a f (x) dx means the limit of F (b) F (a) for very large b (if this limit exists!). Formal
knowledge of limits is not expected.
For solutions see www.maths.ox.ac.uk/r/matlive
Oxford MAT Livestream 2022 – Algebra
MAT syllabus
Simple simultaneous equations in one or two variables. Solution of simple inequalities. Binomial Theorem with positive whole exponent. Combinations and binomial probabilities.
Revision
• Given two linear equations for x and y like 2x + 3y = 7 and 3x + 5y = 9, solve
by rearranging the first for x and substituting that into the second equation, then
rearranging the resulting equation for y, before back-substituting for x.
• We can add and subtract from each side of an inequality.
• We can multiply each side of an inequality by a number, but if we multiply by a negative
number then the direction of the inequality changes. For example, 6 < 3, but when
we multiply both sides by 13 we need to flip the sign to get the true statement 2 > 1.
• Squaring each side of an inequality is like multiplying, and we need to be careful about
cases where that’s positive or negative. For example, 2 < 1, but squaring each
side would be like multiplying both sides by a negative number (so the direction of the
inequality would change). Worse, 1 < 3, and squaring would be like multiplying the
left by a negative number and the right by a positive number; that’s not OK at all!
• (Binomial Theorem) If n is a positive whole number then
✓ ◆
n r n r
n
n
n 1
(x + y) = x + nx y + · · · +
xy
+ · · · + nxy n 1 + y n
r
✓ ◆
n
n!
where
=
and n! means n ⇥ (n 1) ⇥ (n 2) ⇥ · · · ⇥ 2 ⇥ 1 for a whole
r
r!(n r)!
number n.
✓ ◆
n
• (Combinations) Given n di↵erent items, there are
ways to choose r of them.
r
• Proof of the above; suppose we list the n items and take the first r items on the list.
There are n! ways to list the items, but we’ve over-counted the ways to choose r items
because it doesn’t matter what order the first r are in, and it doesn’t matter what order
the other (n r) items are in. The way to fix all this over-counting is to divide; each
real set of r items appears in the above plan r!(n r)! times; the first factorial allows
for all the separate orders of the r items, and the other allows for all the orders of the
n!
(n r) items. So final answer is
.
r!(n r)!
For solutions see www.maths.ox.ac.uk/r/matlive
Oxford MAT Livestream 2022 – Algebra
• If the order of the items chosen matters (e.g. if we’re counting the number of ways to
choose who wins a prize, then who comes second, and so on) then we should only allow
for over-counting from the irrelevant orderings of the (n r) other items, giving the
n!
final answer
.
(n r)!
• (Binomial probabilities) If n independent events each have probability p of success and
probability
q of failure, with p + q = 1, then the probability of exactly r successes is
✓ ◆
n r n r
pq
for 0 6 r 6 n.
r
Warm-up
1. Solve the simultaneous equations x + 4y = 1 and 2x
y = 3.
2. Solve x2 + 2x + xy + y 2 = 5 and x + y = 2.
3. Solve x2 + y = 1 and x + y 2 = 1.
4. For which values of x is it true that x2 + 4x + 3 > 0?
5. Expand (2x + 3y)3
6. Given
2 < a < 1, what can you say about a2 ?
7. Given a < b and c < d, what (if anything) can you say about the relationship between
ac and bd? What (if anything) can you say if you’re also told that a > 0 and c > 0?
8. I’m going to flip five fair coins (each is heads or tails with equal probability, and each
is independent of the others). What’s the probability that I get exactly three heads?
✓ ◆
n r n r
9. Check that the binomial probabilities
pq
for 0 6 r 6 n add up to 1.
r
10. I’ve got six cards that have the numbers one to six on them. I’m going to shu✏e them
and then deal them out from left to right.
• What’s the probability that the cards alternate between odd and even numbers
(either starting with an odd number or an even number, then switching between
odd and even for each subsequent card)?
• What’s the probability that the first three cards I deal out are all prime numbers?
• What’s the probability that the first two cards I deal out are both prime numbers?
• What’s the probability that the first five cards I deal out are all prime numbers?
For solutions see www.maths.ox.ac.uk/r/matlive
Oxford MAT Livestream 2022 – Polynomials
MAT syllabus
The quadratic formula. Completing the square. Discriminant. Factorisation. Factor Theorem.
Revision
• The discriminant of a quadratic ax2 + bx + c = 0 is b2 4ac. If the discriminant is
positive then the quadratic has two real solutions. If the discriminant is zero then
there’s one (repeated) real solution. If the discriminant is negative then there are no
real solutions.
p
b ± b2 4ac
2
2
• If b
4ac > 0, then the solution(s) of ax + bx + c = 0 are x =
.
2a
• ax2 + bx + c can be written as a(x
given by the quadratic formula.
↵)(x
4ac > 0, where ↵ and
) if b2
are roots
• (Complete the square) We can write x2 + bx + c in the form (x + r)2 + p because
2
x + bx + c =
✓
b
x+
2
◆2
✓
+ c
b2
4
◆
.
This is handy if we’re trying to prove that the quadratic is non-negative, because
anything squared is non-negative.
• (Di↵erence of two squares) The expression x2
comes up quite a lot!
a2 factorises as (x
• (Factor Theorem) If p(a) = 0 for a polynomial p(x), then (x
a)(x + a). This
a) is a factor of p(x).
• The degree of a polynomial is the highest power of x, so the degree of any quadratic is
2, and the degree of any cubic is 3, for example.
• When sketching the graph of y = ax2 + bx + c, we need to consider whether a is positive
or negative (whether it’s a “happy” or “sad” quadratic), whether the quadratic has any
roots, and where it crosses the y-axis.
• Sometimes a function which is not a quadratic might secretly be a quadratic in a
di↵erent variable. For example, y = e2x + ex+3 1 is not a quadratic, but if we write
u = ex then we have y = u2 + e3 u 1, which is a quadratic. This is sometimes called
“changing variable”.
For solutions see www.maths.ox.ac.uk/r/matlive
Oxford MAT Livestream 2022 – Primes and Proof
MAT syllabus
[There is no content on primes or proof explicitly on the MAT syllabus, but candidates are
expected to know material from GCSE or equivalent.]
Revision
• A whole number d is a “factor” (or “divisor”) of another whole number n if there exists
a whole number k with n = dk. If this is the case, then we say that “d divides n”.
• A prime number is a whole number greater than 1 which has no factors except for 1
and itself.
• A rational number is a number that can be written as p/q where p and q are whole
numbers. An irrational number is a number that is not rational.
• It’s sometimes a good idea to prove a statement by showing that if it’s not true then
ln 2
nonsense follows as a result. For example, here is a proof that
is irrational.
ln 3
ln 2
Suppose that
is a rational number.
ln 3
ln 2
p
Then
= for some whole numbers p > 0 and q > 0.
ln 3
q
This rearranges to q ln 2 = p ln 3 so 2q = 3p .
But the left-hand side is even and the right-hand side is odd.
ln 2
So the supposition in the first line is false, and
is not a rational number.
ln 3
• If a whole number n is even, then we can write it as n = 2m for some whole number
m. If n is odd, then we can write it as n = 2m + 1. Similarly, all numbers can either
be written as 3m or 3m + 1 or 3m + 2 for some whole number m. Sometimes checking
these di↵erent cases can help us to prove something.
• A “counterexample” to a claim is an example that demonstrates that the claim is not
true. For example, if I make the claim “all square numbers are odd” then you could give
42 = 16 as a counter-example, but 52 = 25 and sin(30 ) = 12 are not counter-examples.
• Sometimes we’re asked to show that something (call it P ) is “sufficient” for something
else (call it Q). We can show that P is sufficient for Q by showing that if P is true,
then Q is also true.
• Sometimes we’re asked to show that something (call it P ) is “necessary” for something
else (call it Q). We can show that P is necessary for Q by showing that if Q is true,
then P is also true.
For solutions see www.maths.ox.ac.uk/r/matlive
Oxford MAT Livestream 2022 – Recursion
MAT syllabus
[There is no content on recursion explicitly on the MAT syllabus, but candidates are expected
to be able to adapt their knowledge of sequences and series to more complex situations.]
Revision
See sequences and series.
• We might use the notation ai,j to refer to a sort of function that depends on two whole
numbers i and j, perhaps with some restrictions on the values that i and j can take,
such as 0 < j 6 i. It might be convenient to imagine this as a table of values
i
1
2
3
..
.
1
a1,1
a2,1
a3,1
..
.
j
2
3
...
a2,2
a3,2
..
.
a3,3
..
.
..
.
• We might write a function with two inputs as f (x, y) or fy (x) or fx (y) depending on
the context (but be consistent!)
Warm-up
1. A sequence is defined by a0 = 3 and then for n > 1 an is the sum of all previous terms.
Find an in terms of n for n > 1.
2. The Fibonacci numbers are defined by F0 = 1 and F1 = 1 and then for n > 2,
Fn = Fn 1 + Fn 2 . Find F10 .
3. Suppose that we’re trying to form a discussion group of r people from a wider collection
of n people, one of whom is Sophie Germain. We’ll write f (n, r) for the number of
di↵erent groups we can form. By considering separate cases for discussion groups that
do or do not contain Sophie Germain, explain why
f (n, r) = f (n
1, r) + f (n
1, r
1)
4. A sequence is defined by C0 = 1 and then for n > 0,
Cn+1 =
n
X
Ci Cn i .
i=0
Find C1 and C2 and C3 and C4 .
For solutions see www.maths.ox.ac.uk/r/matlive
Oxford MAT Livestream 2022 – Geometry
MAT syllabus
Co-ordinate geometry and vectors in the plane. The equations of straight lines and circles.
Basic properties of circles. Lengths of arcs of circles.
Revision
• Points in the plane can be described with two co-ordinates (x, y). The x-axis is the line
y = 0, and the y-axis is the line x = 0.
✓ ◆
x
• A vector
can store the same information as a pair of co-ordinates. Used in that
y
sense, the vector is called a position vector.
✓ ◆
2
• A vector can also describe the displacement from one point to another, so that
1
could represent the displacement from (1, 1) to (3, 2) for example.
• Vectors can be added by adding the components separately. To show that in a diagram,
we might interpret the first vector as a position vector (drawing an arrow starting from
the origin) and then interpret the second as a displacement (drawing an arrow starting
from the end of the first vector).
✓ ◆
p
x
• The magnitude of the vector
is x2 + y 2 .
y
• The distance from A to B is the magnitude
p of the vector displacement from A to B.
The distance from (x1 , y1 ) to (x2 , y2 ) is (x2 x1 )2 + (y2 y1 )2 .
• A vector can be multiplied by a number by multiplying each component by that number.
The result is a vector in the same direction but with scaled magnitude.
• A straight line has equation y = mx+c, where m is the gradient and c is the y-intercept.
Other ways to write the equation of a line are ax + by + c = 0 (where that’s a di↵erent
c to the one in the previous expression) or y y1 = m(x x1 ). The last expression
is useful because that line goes through the point (x1 , y1 ) and has gradient m, which
might be information that we’ve been given.
• Two lines are parallel if and only if they have the same gradient. Two lines are perpendicular if and only if their gradients multiply to give 1.
• The equation of a circle with centre (a, b) and radius r is (x
a)2 + (y
b)2 = r2 .
• The angle in a semicircle is a right angle; if AB is the diameter of a circle, and C is on
the circle, then \ACB = 90 .
For solutions see www.maths.ox.ac.uk/r/matlive
Oxford MAT Livestream 2022 – Geometry
• The tangent is at right angles to the radius at any point on a circle’s circumference.
• A circle with radius r has area ⇡r2 and circumference 2⇡r.
• If two radii of a circle of radius r make an angle of ✓ (in degrees), then the length of
✓
the arc between those radii is
2⇡r. The area of the sector enclosed by that arc and
360
✓
the radii is
⇡r2 .
360
Warm-up
✓ ◆
✓ ◆
✓ ◆
3
4
1
1. Draw a diagram to show the three separate position vectors
and
and
.
2
1
2
✓ ◆
✓ ◆
3
4
2. Add the vectors
and
. Show this on your diagram.
2
1
✓ ◆
✓ ◆
4
1
3. Find 3
+2
. Show this on your diagram.
1
2
4. Find the equation of the line through the points (1, 5) and (3, 1).
5. Find the equation of the line through the point (3, 5) with gradient 2.
6. Find equations of three lines such that the region bounded by the three lines is an
equilateral triangle.
7. A circle has centre ( 1, 4) and radius 3. Write down an equation for the circle. What’s
the area of this circle?
8. A circle is given by x2 + 9x + y 2
3y = 10. Find the centre and radius of the circle.
9. Points A and B lie on a circle with centre O and radius 2. The angle \AOB is 120 .
Find the length of the arc between A and B. Find the area enclosed by that arc and
the radii OA and OB.
10. A circle is given by x2 + y 2 = 4. The line x = 1 splits the circle into two regions. Find
the area of each region.
11. Two circles are given by x2 + y 2 = 4 and (x
that’s inside both circles.
2)2 + y 2 = 4. Find the area of the region
12. The points (0, 0) and (1, a) and (0, a + 1/a) all lie on the same circle. Find the centre
of the circle in terms of a.
Hint: angles.
For solutions see www.maths.ox.ac.uk/r/matlive
Oxford MAT Livestream 2022 – Trigonometry
MAT syllabus
Solution of simple trigonometric equations. The identities
sin x
, sin2 x + cos2 x = 1, sin (90
x) = cos x.
cos x
Periodicity of sine, cosine and tangent. Sine and cosine rules for triangles.
tan x =
Revision
B
C
✓
A
• If triangle ABC is right-angled at A and \BCA = ✓, then we define
sin ✓ =
• So tan ✓ =
|AB|
,
|BC|
cos ✓ =
|AC|
,
|BC|
tan ✓ =
|AB|
.
|AC|
sin ✓
.
cos ✓
• Also Pythagoras’ Theorem states that |AB|2 + |AC|2 = |BC|2 so sin2 ✓ + cos2 ✓ = 1. It
follows that 1 6 sin ✓ 6 1 and 1 6 cos ✓ 6 1.
• Since the angles in a triangle add up to 180 , the angle at B is 90
Looking at the triangle that way around, we can deduce that sin(90
cos(90
✓) = sin ✓.
✓.
✓) = cos ✓ and
• The sine, cosine, and tangent functions are all periodic;
sin(x + 360 ) = sin x,
• Also, we have sin( x) =
cos(x + 360 ) = cos x,
tan(x + 180 ) = tan x.
sin x, and cos( x) = cos(x).
Now consider a triangle that is not necessarily right-angled.
B
a
C
c
↵
b
A
For solutions see www.maths.ox.ac.uk/r/matlive
Oxford MAT Livestream 2022 – Trigonometry
respectively, and call the side-lengths a = |BC|,
Let’s write the angles at A, B, C as ↵, ,
b = |AC|, c = |AB|.
• The area of this triangle is
1
ab sin
2
1
1
= bc sin ↵ = ca sin .
2
2
• (The cosine rule) We have a2 = b2 + c2
• (The sine rule) We have
sin ↵
sin
=
a
b
2bc cos ↵.
=
sin
.
c
Warm-up
1. Write down the values of sin x and cos x and (where possible) tan x when x is equal to
0 , 30 , 45 , 60 , 90 , 120 , 135 , 150 , 180 , 210 , 225 , 240 , 270 , 300 , 315 , 330 .
2. Find all the solutions to sin x =
1
with 0 6 x < 360 .
2
3. Find all the solutions to tan x = 1 with 0 6 x < 360 .
4. Find all the solutions to tan(45x) = 1 with 0 6 x < 360 . Careful, not just 1 and 5 .
5. Write (cos x + sin x)2 in terms of the variable u = cos x sin x.
6. For 0 6 x < 90 , write 1 sin2 x + sin4 x sin6 x + . . . as a single expression (not an
infinite sum) in terms of cos x. Why have I excluded 90 from the range here?
7. Write cos4 x + cos2 x in terms of sin x.
8. Simplify cos(450
9. Simplify cos(90
x).
x) sin(180
If you know a fact about sin(A
x)
sin(90
x) cos(180
x).
B), you may only use it here if you prove it!
10. A triangle ABC has side lengths AB = 3 and BC = 2, and the angle \ABC = 120 .
Find the remaining side length AC, the area of the triangle, and an expression for
sin \BCA.
11. From the cosine rule, and the fact that 1 6 cos ↵ 6 1, deduce that for any triangle
with side lengths a > b > c > 0, we must have a < b + c (this is the rule that “the
longest side is shorter than the sum of the other two sides” or said di↵erently “the
shortest route from B to C is a straight line” also known as “the triangle inequality”).
For solutions see www.maths.ox.ac.uk/r/matlive
Oxford MAT Livestream 2022 – Sequences and series
MAT syllabus
Sequences defined iteratively and by formulae. Arithmetic and geometric progressions*.
Their sums*. Convergence condition for infinite geometric progressions*.
* Part of full A-level Mathematics syllabus.
Revision
• A sequence an might be defined by a formula for the nth term like an = n2
n.
• A sequence an might be defined with an relation like an+1 = f (an ) for n > 0, if we’re
given the function f (x) and also given a first term like a0 = 1. (The “first term” might
be a0 if we feel like counting from zero).
• The sum of the first n terms of a sequence ak can be written with the notation
(if the first term is a0 ) or
n
X
n 1
X
ak
k=0
ak (if the first term is a1 ).
k=1
• An arithmetic sequence is one where the di↵erence between terms is constant. The
terms can be written as a, a + d, a + 2d, a + 3d, . . . , where a is the first term and d is
the common di↵erence.
• The sum of the first n terms of an arithmetic sequence with first term a and common
n
di↵erence d is (2a + (n 1)d), which you can remember as “first term plus last term,
2
times the number of terms, divided by two”.
• A geometric sequence is one where the ratio between consecutive terms is constant.
The terms can be written as a, ar, ar2 , ar3 , . . . where a is the first term and r is the
common ratio.
• The sum of the first n terms of a geometric sequence with first term a and common
a(1 rn )
ratio r is
. One way to remember this is to remember what happens if we
1 r
multiply the sum of the first n terms of a geometric series by (1 r),
(1
r)(a + ar + · · · + arn 1 ) =(a ar) + (ar
=a arn .
ar2 ) + · · · + (arn
• For a geometric sequence an , the sum to infinity is written as
a
1
X
1
arn )
ak . If the common
k=0
. If |r| > 1 then this sum to infinity
1 r
does not converge (it does not approach any particular real number).
ratio r satisfies |r| < 1 then this is equal to
For solutions see www.maths.ox.ac.uk/r/matlive