0580 · 2.7Sequences

Sequence rules and notation

0:00A row of joined squares uses four matchsticks, then seven, then ten. Each new square shares one side with the previous square. That means extending the row needs only three extra matchsticks. A sequence records this growing pattern, and an algebraic rule lets us predict a distant pattern without drawing every square. By the end of this video, you will recognise different sequence patterns and continue them correctly. You will also find rules connecting a term to its position.

0:25We will explore linear, quadratic, cubic and exponential sequences, including simple combinations. The aim is to understand what makes each type behave differently. You need to be comfortable with subtraction, substitution and powers. Squaring a number means multiplying it by itself, while cubing means raising it to the power of three. Substitution means replacing a letter with a specified number. We will use these familiar operations to turn patterns into rules.

0:52A sequence is an ordered list of terms following a rule. A term is one of the numbers or objects in that list. Its position tells us where it appears, starting with position one unless stated otherwise. In our matchstick sequence, the fourth term is thirteen, but its position is four. The notation T subscript n means the nth term of sequence T. The letter n represents the position, not the value of the term.

1:15For example, T subscript four means the term in position four. An nth term rule connects any allowed position directly to its corresponding value. There are two important ways to describe a sequence. A term-to-term rule tells us how to move from one term to the next. For the matchsticks, that rule is add three. A position-to-term rule tells us how to calculate a term directly from its position. The matchstick pattern starts with four sticks, then adds three for each additional square.

1:43At position n, there have been n − 1 additions. The total is therefore 4 + 3 times the quantity n − 1. Simplifying gives 3n + 1, so the hundredth pattern uses three hundred and one matchsticks. A term-to-term rule needs a starting term to identify the sequence. The instruction add three could describe many different lists. An nth term rule gives a direct route to a distant term without calculating all the earlier ones.

2:07Always identify which kind of rule a question asks you to give.

Linear sequences and nth terms

2:11To continue a sequence, look for a consistent pattern across all the supplied terms. For twenty, fifteen, ten and five, subtracting five gives zero, then negative five. For one, three, six and ten, the additions increase from two to three to four. The next additions are five and six, giving fifteen and twenty-one. Differences are found by subtracting each term from the following term. They help reveal patterns that are not obvious in the original list.

2:36A short list can mathematically fit more than one rule, so consider every supplied term and any diagram. In examination questions, look for the simple consistent pattern supported by that information. Our first worked example asks us to find the nth term of seven, eleven, fifteen and nineteen. Subtracting consecutive terms gives four each time. The sequence 4n produces four, eight, twelve and sixteen. Each given term is three greater than the matching term of 4n.

3:05The nth term is therefore 4n + 3. For the fiftieth term, substitute fifty for n. 4 × 50 gives two hundred. Adding three gives two hundred and three, without needing to list the previous forty-nine terms. A sequence with constant first differences is linear. Its nth term has the form a times n + b. The number a is the common difference because increasing n by one increases a times n by a.

3:31The number b adjusts that sequence so its terms match the supplied values. The first term is not automatically the constant added at the end. In this example, the first term is seven, but the adjustment is three. That is because 4n already contributes four when n = 1. Checking your rule at positions one, two and three helps catch this common mistake. We can also use the rule to test whether a number belongs to the sequence.

3:55For ninety-nine, set 4n + 3 equal to ninety-nine. Subtracting three gives 4n equal to ninety-six. Dividing by four gives n equal to twenty-four, so ninety-nine is the twenty-fourth term. For one hundred, the same calculation gives 97 ÷ 4, which is twenty-four point two five. That is not a whole-number position in this sequence. One hundred is therefore not a term. A solution must be a positive integer when the positions begin at one.

Quadratic sequences and second differences

4:24Quadratic sequences behave differently because their first differences are not constant. Consider the square numbers one, four, nine, sixteen and twenty-five. Their first differences are three, five, seven and nine. Subtracting again gives constant second differences of two. A growing square provides a reason for this pattern. Increasing an n by n square to the next size adds an extra row and column. Together, these contain 2n + 1 new unit squares.

4:51That addition increases by two each time, explaining the constant second difference. A quadratic nth term has the form a times n squared plus b times n plus c. The n² part creates the constant second difference. The linear and constant parts do not change it. Since n² has second difference two, multiplying it by a makes the second difference two times a. Our second worked example asks for the nth term of five, twelve, twenty-three, thirty-eight and fifty-seven.

5:20The first differences are seven, eleven, fifteen and nineteen. Their differences are four, four and four. Dividing four by two gives two, so we begin with 2n². At the first five positions, 2n² gives two, eight, eighteen, thirty-two and fifty. Subtracting these from the supplied terms leaves three, four, five, six and seven. That remainder has the linear rule n + 2. Adding it back gives 2n² + n + 2.

5:51Check position three by calculating 2 × 3², then adding three and two. The square is nine, so this becomes 18 + 3 + 2. The result is twenty-three, matching the original sequence. For position ten, the expression becomes two times one hundred plus ten plus two, giving two hundred and twelve. We can continue the same sequence using its differences. Adding four to the last first difference, nineteen, gives twenty-three.

6:18Adding twenty-three to fifty-seven gives the next term, eighty. The difference method and the nth term rule describe the same pattern from different viewpoints. Do not use the second difference itself as the coefficient of n². The coefficient is half that difference because square numbers already have second difference two. Also, finding that coefficient does not finish the rule. Subtract its contribution to identify the remaining linear part.

Cubic sequences and third differences

6:44Cubic sequences involve n³ rather than n². The cube numbers are one, eight, twenty-seven, sixty-four and one hundred and twenty-five. Their first differences are seven, nineteen, thirty-seven and sixty-one. Taking differences twice more produces constant third differences of six. Constant third differences that are not zero indicate a cubic sequence. For simple examples, compare the terms with familiar cubes and look at what remains. A multiple of n³ produces the same multiple of six as its third difference.

7:13Lower-degree additions can change earlier differences without changing that third difference. Our third worked example uses two, ten, thirty, sixty-eight and one hundred and thirty. The first differences are eight, twenty, thirty-eight and sixty-two. The second differences are twelve, eighteen and twenty-four. Their differences are six and six, suggesting we compare the sequence with n³. Subtracting the matching cube numbers leaves one, two, three, four and five. That remainder is exactly the position number, n.

7:43The nth term is therefore n³ + n. For the sixth term, 6³ gives two hundred and sixteen, then adding six gives two hundred and twenty-two. These examples show how relationships between sequences help us find rules. A complicated-looking sequence may be a familiar sequence with something added or subtracted. Compare matching positions, rather than unrelated terms. Always verify the combined rule against the original list before using it for another position.

Exponential sequences and exam skills

8:09Our fourth worked example uses three, six, twelve and twenty-four. Dividing each term by the previous one gives a constant ratio of two. The term-to-term rule is multiply by two. This is an exponential sequence because the position appears in the exponent of its nth term. The starting value is three, and reaching position n requires n − 1 doublings. The rule is three multiplied by two to the power of n minus one, where the whole of n − 1 is the power.

8:36At position one, the exponent is zero, giving 3 × 1. For position six, 2⁵ is thirty-two, and multiplying by three gives ninety-six. The exponent is n − 1 because no doubling has happened at the first term. Using n instead would already double the starting value. Exponential sequences can also decrease, such as repeatedly multiplying by 1/2. Recognise the constant multiplier rather than assuming every sequence increases by addition.

9:04A simple combination might be three, five, nine, seventeen and thirty-three. Its consecutive ratios are not constant, so multiplying the whole term by two does not work. Subtracting one from each term reveals two, four, eight, sixteen and thirty-two. Those are powers of two, so the rule is 2ⁿ, then add one. For the sixth term, 2⁶ gives sixty-four. Adding one gives sixty-five. Notice that the extra one sits outside the power.

9:34If the plus one were inside the power, the rule would describe a different sequence, so written grouping matters. In an examination, you may be asked to continue a sequence, describe a term-to-term rule, or find and use the nth term. You might also count a growing diagram or compare two related sequences. Show a difference table when it supports your reasoning. Check the final rule by substituting early positions before answering any later parts.

9:59These methods must work without a calculator on Paper two, and they are also useful on calculator Paper four. Practise subtraction, small powers and substitution by hand. Keep these integer answers exact rather than rounding them to three significant figures. If a different calculation requires rounding, retain full working values until the final answer. A sequence connects position with value through a consistent pattern. Constant first, second and third differences identify linear, quadratic and cubic sequences respectively.

10:28A constant multiplier identifies a basic exponential sequence. Comparing with familiar sequences can reveal simple combinations and turn an unfamiliar list into an understandable rule.