Tag Archives: number sequences

Number Sequences

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Number sequences appear in Nature all over the place, from sunflowers to conch shells. They can also crop up either in Maths or Verbal Reasoning, and both are essential parts of 11+ and other school examinations.

The trick is to be able to recognise the most common sequences and, if you find a different one, to work out the pattern so that you can find the missing values (or ‘terms’).

Common Sequences

Here are a few of the commonest number sequences. For each one, I’ve given the rule for working out the nth term, where n stands for its position in the sequence.

Even numbers: 2, 4, 6, 8 etc… Rule: 2n
Odd numbers: 1, 3, 5, 7 etc… Rule: 2n – 1
Powers of 2: 2, 4, 8, 16 etc… Rule: 2ⁿ
Prime numbers: 2, 3, 5, 7 etc… Rule: n/a (each number is only divisible by itself and one)
Square numbers: 1, 4, 9, 16 etc… Rule: n²
Triangular numbers: 1, 3, 6, 10 etc… Rule: sum of the numbers from 1 to n
Fibonacci sequence: 1, 1, 2, 3 etc… Rule: n₋₂ + n₋₁ (ie each successive number is produced by adding the previous two numbers together, eg 1 + 1 = 2, 1 + 2 = 3)
Factors: 36, 18, 9, 6, 4 etc Rule: Factors from largest to smallest (tricky one, this…!)

Here are a few questions for you to try. What are the next two numbers in each of the following sequences?

  1. 14, 16, 18, 20…
  2. 9, 16, 25, 36…
  3. 3, 6, 12, 24…
  4. 7, 11, 13, 17…
  5. 5, 8, 13, 21…

Working out the Pattern

The best way to approach an unfamiliar sequence is to calculate the gaps between the terms. Most sequences involve adding or subtracting a specific number, eg 4 in the case of 5, 9, 13, 17 etc.

Sometimes, the difference will rise or fall, as in 1, 2, 4, 7 etc. If you draw a loop between each pair of numbers and write down the gaps (eg +1 or -2), the pattern should become obvious, enabling you to work out the missing terms.

  • If the missing terms are in the middle of the sequence, you can still work out the pattern by using whatever terms lie next to each other, eg 1, …, 5, 7, …, 11 etc. You can confirm it by checking that the gap between every other term is double that between the ones next to each other.
  • If the gaps between terms are not the same and don’t go up (or down) by one each time, it may be that you have to multiply or divide each term by a certain number to find the next one, eg 16, 8, 4, 2 etc.
  • If the gaps go up and down, there may be two sequences mixed together, which means you’ll have to look at every other term to spot the pattern, eg 1, 10, 2, 8 etc. Here, every odd term goes up by 1 and every even term falls by two.

Generating a Formula

At more advanced levels, you may be asked to provide the formula for a number sequence.

Arithmetic Sequences

If the gap between the terms is the same, the sequence is ‘arithmetic’. The formula for the nth term of an arithmetic sequence is xn ± k, where x is the gap, n is the position of the term in the sequence and k is a constant that is added or subtracted to make sure the sequence starts with the right number, eg the formula for 5, 8, 11, 14 etc is 3n + 2.

The gap between each term is 3, which means you have to multiply n by 3 each time and add 2 to get the right term, eg for the first term, n = 1, so 3n would be 3, but it should be 5, so you have to add 2 to it. Working out the formula for a sequence is particularly useful at 13+ or GCSE level, when you might be given a drawing of the first few patterns in a sequence and asked to predict, say, the number of squares in the 50th pattern.

You can also work out the position of the pattern in the sequence if you are given the number of elements. You do this by rearranging the formula, ie by adding or subtracting k to the number of elements and dividing by 𝒳. For example, if 3n +2 is the formula for the number of squares in a tiling pattern, and you have 50 squares in a particular pattern, the number of that pattern in the sequence = (50-2) ÷ 3 = 48 ÷ 3 = 16.

Quadratic Sequences

If the gap between the terms changes by the same amount each time, the sequence is ‘quadratic’, which just means there is a square number involved.

The formula for a quadratic sequence is 𝒳n² ± k, where 𝒳 is half the difference between the gaps (or ‘second difference’), n is the position of the term in the sequence and k is a constant that is added or subtracted to make sure the sequence starts with the right number, eg the formula for 3, 9, 19, 33 etc is 2n² + 1.

The differences between the terms are 6, 10, 14, so the second difference is 4, which means you need to multiply the square of n by 4 ÷ 2 = 2 and add 1, eg for the first term, n = 1, so 2n² would be 2, but it should be 3, so you have to add 1 to it.

Geometric Sequences

If each term is calculated by multiplying the previous term by the same number each time, the sequence is ‘geometric’. The formula for the nth term of a geometric sequence (or progression) is ar(n-1), where a is the first term, r is the multiplier (or ‘common ratio’) and n is the position of the term in the sequence, eg the formula for 2, 8, 32, 128 etc is 2 x 4(n-1). The first term is 2, and each term is a power of 4 multiplied by 2, eg the fourth term = 2 x 4(4-1) = 2 x 43 = 2 x 64 = 128.

Here are a few questions for you to try. What is the formula for the nth term in each of the following sequences?

  1. 14, 16, 18, 20…
  2. -1, 3, 7, 11…
  3. 4, 6, 10, 16…
  4. 9, 7, 5, 3…
  5. 2, 6, 18, 54… 

     

     

     

    If you’re looking for past papers with answers, especially in the run-up to 11+/13+ exams, GCSEs or A-levels, you can visit my Past Papers page and subscribe for just £37.99 a year.

Useful Formulas

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What is a problem? A problem = a fact + a judgment. That is a simple formula that tells us something about the way the world works. Maths is full of formulas, and that can intimidate some people if they don’t understand them or can’t remember the right one to use.

However, formulas should be our friends as they help us to do complex calculations accurately and repeatably in a consistent and straightforward way. The following is a list of the most useful ones I’ve come across while teaching Maths to a variety of students at a variety of ages and at a variety of stages in their education.

Averages

  • The mean is found by adding up all the values and dividing the total by how many there are, eg the mean of the numbers 1-10 is 5.5, as 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55, and 55 ÷ 10 = 5.5.
  • The mode is the most common value (or values), eg the mode of 1, 2, 2, 3, 4, 5 is 2.
  • The median of an odd number of values sorted by size is the one in the middle, eg the median of the numbers 1-5 is 3. The median of an even number of values is the mean of the two numbers in the middle, eg the median of the numbers 1-10 is 5.5, as 5 and 6 are the numbers in the middle, and 11 ÷ 2 = 5.5.
  • The range is the highest value minus the lowest, eg the range of the numbers 1-10 = 10 – 1 = 9.

Sample Questions

  1. What is the mean of the numbers 6, 5, 8, 2 and 4?
  2. What is the mode of the numbers 1, 5, 5, 12 and 3?
  3. What is the median of the numbers 2, 8, 9, 6 and 5?
  4. What is the median of the numbers 1, 13, 4, 6, 8 and 20?
  5. What is the range of the numbers 15, 2, 3, 8 and 4?

Circle Theorems

  • Alternate segment theorem
    The angle between a tangent and a chord is equal to the angle subtended by the same chord in the alternate segment.
  • Angles at the centre and at the circumference
    The angle formed at the centre of the circle by lines starting from two points on the circumference is double the angle formed on the other side of the circle by lines originating from the same points.
  • Angles in the same segment
    If you create two triangles between the ends of the chord and the side of the circle, the angles in the same segment are equal.
  • Angles in a semicircle
    If you draw a triangle using the diameter and any point on the circumference, the angle at that point is 90°.
  • Perpendicular chord bisection
    The perpendicular from the centre of a circle to a chord will always bisect the chord (divide it into two equal parts).
  • Tangent of a circle
    a) The angle between a tangent and radius is 90°.
    b) Tangents that meet at the same point outside the circle are equal in length.
  • Cyclic quadrilateral
    The opposite angles in a cyclic quadrilateral add up to 180°.
  • Angles subtended on the same arc
    Angles formed from two points on the circumference of a circle are equal to other angles, in the same arc, formed from those two points.
  • Angle in a semi-circle
    Angles created by drawing lines from the ends of the diameter of a circle to the circumference form a right angle.

Geometry

  • Angles around a point add up to 360º
  • Angles on a straight line add up to 180º
  • Vertically opposite angles are equal, ie the two pairs of angles opposite each other when two straight lines bisect (or cross) each other
  • Alternate angles are equal, ie the angles under the arms of a ‘Z’ formed by a line (or ‘transversal’) bisecting two parallel lines
  • Corresponding angles are equal, ie the angles under the arms of an ‘F’ formed by a line (or ‘transversal’) bisecting two parallel lines
  • Complementary angles add up to 90º
  • Any straight line can be drawn using y = mx + c, where m is the gradient and c is the point where the line crosses the y-axis (the ‘y-intercept’)
  • The gradient of a straight line is shown by δy/δx (ie the difference in the y-values divided by the difference in the x-values of any two points on the line, usually found by drawing a triangle underneath it)

Polygons

  • Number of diagonals in a polygon = (n-3)(n÷2) where n is the number of sides
  • The sum of the internal angles of a polygon = (n-2)180º, where n is the number of sides
  • Any internal angle of a regular polygon = (n-2)180º ÷ n, where n is the number of sides

Sample Questions

  1. How many diagonals are there in a square?
  2. How many diagonals are there in an octagon?
  3. What is the sum of the internal angles of a pentagon?
  4. What is the sum of the internal angles of a hexagon?
  5. What are the internal angles of a regular decagon?
  6. What are the internal angles of a regular heptagon?

Probability

  • The probability of something happening = the number of ways of getting what you want divided by the number of possible outcomes, eg the chance of getting heads if you flip a coin is 1/2 (or 0.5 or 50%).
  • The probability of a sequence of events = the probability of all the events multiplied together, eg the chance of getting two heads in a row if you flip a coin is 1/2 x 1/2 = 1/4 (or 0.25 or 25%).

Sample Questions

  1. What is the probability of rolling a six?
  2. If there are two red and three black balls in a bag, what is the probability of taking out a red one?
  3. What is the probability of rolling a number greater than 3?
  4. What is the probability of tossing three heads in a row?
  5. If the chance of hitting treble 20 is 1/10, what is the probability of scoring 180 with three darts?

Rectangles

  • Perimeter of a rectangle = 2(l + w), where l = length and w = width
    Note that this is the same formula for the perimeter of an L-shape, too.
  • Area of a rectangle = lw, where l = length and w = width

Sample Questions

  1. What is the perimeter of a rectangle measuring 5 x 7 cm?
  2. What is the perimeter of a rectangle measuring 13 x 4 cm?
  3. What is the perimeter of an L-shape measuring 12 x 8 m overall with a 4 x 2 m piece missing?
  4. What is the area of a rectangle measuring 3 x 12 cm?
  5. What is the area of a rectangle measuring 6 x 8 cm?

Trapeziums

  • Area of a trapezium = lw, where l = average length and w = width (in other words, you have to add both lengths together and divide by two to find the average length)

Sample Questions

  1. What is the area of a trapezium with a height of 7 cm and horizontal sides of 4 and 8 cm?
  2. What is the area of a trapezium with a height of 10 inches and horizontal sides of 5 and 4 inches?
  3. What is the area of a trapezium with a height of 5 cm and horizontal sides of 2 and 4 cm?
  4. What is the area of a trapezium with a height of 12 m and horizontal sides of 6 and 9 m?
  5. What is the area of a trapezium with a height of 10 cm and horizontal sides of 4 and 5 cm?

Triangles (Trigonometry)

  • Area of a triangle = ½bh, where b = base and h = height
  • Angles in a triangle add up to 180º
  • Pythagoras’s Theorem: in a right-angled triangle, a² + b² = c², ie the area of a square on the hypotenuse (or longest side) is equal to the sum of the areas of squares on the other two sides

 

 

 

 

 

 

Circles

  • Circumference of a circle = 2πr, where r = radius
  • Area of a circle = πr², where r = radius (π = 3.14 to two decimal places and is sometimes given as 22/7)

Sample Questions

  1. What is the area of a triangle with a height of 5 cm and a base of 3 cm?
  2. What is the remaining internal angle of a triangle if the others are 20º and 30º?
  3. What is the length of the remaining side of a right-angled triangle if the others are 4 cm and 3 cm?
  4. What is the circumference of a circle with a radius of 5cm, assuming π = 3.14?
  5. What is the area of a circle with a diameter of 4 m, assuming π = 22/7?

Spheres

  • Surface area of a sphere = 4πr², where r = radius
  • Volume of a sphere = 4/3πr³, where r = radius

Sample Questions

  1. What is the surface area of a sphere with a radius of 15 cm, assuming π = 3.14?
  2. What is the surface area of a sphere with a diameter of 7 cm, assuming π = 3.14?
  3. What is the volume of a sphere with a radius of 1 cm, assuming π = 22/7?
  4. What is the volume of a sphere with a radius of 5 cm, assuming π = 3.14?
  5. What is the volume of a sphere with a diameter of 5 m, assuming π = 22/7?

Cuboids

  • Surface area of a cuboid = 2(lw + lh + wh), where l is length, w is width and h is height
  • Volume of a cuboid = lwh, where l is length, w is width and h is height

Sample Questions

  1. What is the surface area of a cuboid measuring 3 x 4 x 5 cm?
  2. What is the surface area of a cuboid measuring 6 x 2 x 12 cm?
  3. What is the volume of a cuboid measuring 8 x 5 x 6 cm
  4. What is the volume of a cuboid measuring 2 x 15 x 7 cm
  5. What is the volume of a cuboid measuring 18 x 9 x 2 cm

Number Sequences

  • The nth term of any arithmetic sequence (with regular intervals) = xn ± k, where x is the interval (or difference) between the values, n is the value’s place in the sequence and k is a constant that is added or subtracted to make sure the sequence starts at the right number, eg the formula for 5, 8, 11, 14…etc is 3n + 2
    • If you know where the term is in the sequence, you can find it by plugging in n and solving the equation, eg the fourth term of the sequence above = 3n + 2 = 3 x 4 + 2 = 14
    • If you know what the term is, you can find its position in the sequence by adding the number as the result of the formula and solving for n, eg if the term in the sequence above = 14, then 3n + 2 = 14, 3n = 12 and n = 4.
  • The sum of n consecutive numbers is n(n + 1)/2, eg the sum of the numbers 1-10 is 10(10 + 1)/2 = 110/2 = 55

Sample Questions

  1. What is the formula for the nth term of 2, 4, 6, 8…?
  2. What is the formula for the nth term of -5, -3, -1, 1…?
  3. What is the seventh term of the sequence 4, 8, 12, 16…?
  4. What position is 66 in the sequence 3, 6, 9, 12…?
  5. What is the sum of the numbers 12 to 18?

Other

  • Speed = distance ÷ time
  • Time = distance ÷ speed
  • Distance = speed x time
  • Profit = sales – cost of goods sold
  • Profit margin = profit ÷ sales
  • Mark-up = profit ÷ cost of goods sold

Sample Questions

  1. What is the speed of a vehicle that travels 20 miles in two hours?
  2. How long does it take to travel 210 km at 70 km/h?
  3. What profit does a butcher make on three steaks he buys at £2 and sells for £2.50?
  4. What’s his profit margin on those steaks?
  5. What’s his mark-up on those steaks? 

     

     

     

    If you’re looking for past papers with answers, especially in the run-up to 11+/13+ exams, GCSEs or A-levels, you can visit my Past Papers page and subscribe for just £37.99 a year.