Finding the nth Term of a Linear Sequence

Picture the stack of chairs at the side of a school hall. The first chair on the floor stands about eighty centimetres tall, and every chair you drop on top lifts the stack by another eight centimetres. Two chairs, eighty-eight centimetres. Three chairs, ninety-six. The caretaker wants to know whether a stack of thirty will fit in the store cupboard, and that ceiling is three metres up. You could add eight over and over until you reach the thirtieth chair, and one slip anywhere along that chain quietly ruins every number after it. What the caretaker actually wants is a rule that takes the number thirty and hands back the height in one step, with no chain to walk at all.

Swap the chairs for a flower bed for a moment, because the numbers there let the two kinds of rule sit side by side where you can compare them. A gardener plants rows of tulip bulbs for a display. Row one has four bulbs, row two has seven, row three has ten, and row four has thirteen. Ask how you get from one row to the next and it is comfortable enough: add three. That is the term-to-term rule, and as an answer to that question it is completely correct. The trouble starts when the question changes: find an expression, in terms of n, for the number of bulbs in row n. Add three is still true about the flower bed, and it is now the answer to a different question. Add three only ever tells you what comes next. To reach row fifty with it you would need row forty-nine first, which means walking every row from the beginning. A position-to-term rule skips that walk. You hand it the row number, and it hands back the bulbs in that row from a standing start. Quick check, and everything you need is already on screen. Three statements about the tulip rows. A, add three to the row before. B, three n plus one. C, row four has thirteen bulbs. Only one of those could give you row fifty in a single step. Pick one. I'll wait. The answer is B, three n plus one. Put fifty in place of n and you get one hundred and fifty-one bulbs, with no row before it needed. A is the chain, and C is one row's value rather than a rule. A and B describe the very same flower bed, and only B answers a question that asks for an expression in n.

Building that expression from four bare numbers takes two steps, and you have already done the first one without noticing. Step one is the gap between the terms. Four to seven is three, seven to ten is three, ten to thirteen is three, so the common difference is three. That three becomes the number sitting in front of n. Here is why that works, and it is worth understanding rather than trusting. A sequence that climbs by three every row is climbing at exactly the same rate as the three times table. So write the three times table underneath it: three, six, nine, twelve, sitting directly under four, seven, ten, thirteen. Two lists climbing at the same rate stay the same distance apart forever, which is why one single number carries you from the times table onto your sequence, and why that number never changes along the way. Step two is finding that number, and you find it by comparing. Three is one below four. Six is one below seven. Nine is one below ten. Every term of the sequence is one more than the three times table. So the nth term is three n plus one, and that is the finished answer, written as an expression in n rather than as a number. Times table, then shift is the whole method in four words. The gap gives you the times table, and one comparison gives you the shift. One warning, because the shift has a convincing lookalike. It is not the first term. Copying the first term across gives three n plus four, and that collapses at once: put n as one and it claims seven bulbs in row one, when row one has four. That test is the habit worth building. Substitute n as one and see whether the first term comes back out, then try one more term to be sure. Now take the method back to the chairs. Eighty, eighty-eight, ninety-six, and the gap is eight every time, so the front of the expression is eight n. The eight times table runs eight, sixteen, twenty-four. Every stack height is seventy-two centimetres above that, so the height of a stack of n chairs is eight n plus seventy-two. Notice the shift came out as seventy-two, not the eighty you started at. Now put thirty in. Eight thirties are two hundred and forty, plus seventy-two, giving three hundred and twelve centimetres. That stack stands twelve centimetres taller than a three metre ceiling, and the caretaker knows it without lifting a single chair.

Time to turn the method on somebody else's answer, because a mistake is far easier to spot in another person's working than in your own. A student is given the sequence six, eleven, sixteen, twenty-one, and asked to find an expression for the nth term. They write plus five. Two jobs for you here. Explain in one sentence why plus five cannot be the answer, and then find the expression that is. Have a go at this one. I'll wait. Plus five is the term-to-term rule. It tells you how to step from one term to the next, and it needs the previous term in your hand before it does anything at all. An expression for the nth term has to work from the position alone. As for the expression, the gap is five, so it starts with five n. The five times table runs five, ten, fifteen, twenty, and every term of the sequence is one more than that, so the answer is five n plus one. One sentence covers the explanation: plus five describes the step between terms, while the nth term has to be a formula built from the position n.

Finding the expression is only half of this skill, and exam questions rarely stop there. The other half is using the thing once you have it. Forwards, it is a substitution and nothing more. For three n plus one, row one hundred holds three hundred and one bulbs, with no rows in between needed. Backwards is the direction that catches people out. A question hands you an nth term and asks whether some particular number appears in that sequence at all. Here is the version to try for yourself. The nth term of a sequence is four n minus three. Is forty-one a term in this sequence? Take your time. I'll wait right here. The answer is yes, and here is the working. If forty-one is a term, then four n minus three equals forty-one for some position n. Add three to both sides, four n equals forty-four, then divide by four, and n comes out as eleven. Eleven is a whole, positive counting number, and that is precisely what makes the answer yes: forty-one is the eleventh term of that sequence. Run the same question with thirty and it goes the other way. Four n comes out as thirty-three, so n is eight point two five, and there is no eight-and-a-quarter-th term, so thirty is not in the sequence. So the deciding question is never whether n comes out at all, it is whether n comes out as a whole positive number, because positions are counted one, two, three, with nothing in between. Two pointers before the exam side. Every sequence here climbed by the same amount each time, and when that gap keeps changing, the video called Finding the nth Term of a Quadratic Sequence takes over. For the total of the first n terms, that is Sum of an Arithmetic Series.

One question on a Foundation paper gave the first few terms of a linear sequence and asked for an expression for the nth term. Here is the examiner's line on what came back. There were many responses where students gave the value of the next term or the ninth term, presumably a misread of nth. Some only gave the term-to-term rule so plus three and n plus three were frequently seen. Those who knew the correct format of the nth term often gave answers such as three n, three n minus four or four n plus three but answers such as three n plus seven and seven n plus three were also common. Plus three and n plus three are both the term-to-term rule, the second one with an n bolted onto the front. Three n on its own has the multiplier right and no shift at all, so the comparison step never happened, and three n minus four shifted the sequence the wrong way. Four n plus three is the two numbers in swapped seats, with the first term promoted to multiplier. Every one of those answers dies under the same test: put n as one and see whether the first term comes back. The wording is doing work as well. In terms of n means your answer must contain the letter n, so a single number can never be a complete answer, however carefully it was worked out.

Times table, then shift. Times table, then shift. Let me put the whole thing back in order before you go. One sequence can be described two ways, and the question tells you which one it wants. The step from each term to the next is the term-to-term rule, and a formula built from the position is the nth term. The common difference is the number in front of n, and that is because your sequence climbs at the same rate as that times table. The shift is whatever moves the times table onto your sequence, and you get it by comparing against a real term, never by copying the first term across. Then check it: put n as one, and the first term should come straight back out at you. To use the expression, substitute a position in to get a term, or set it equal to a value and solve for n, where a whole positive n means that value really is in the sequence. And if you can say in one sentence why plus three on its own does not answer the question, this topic belongs to you.

Next in the chapter: Finding the nth Term of a Quadratic Sequence, where the gaps stop staying the same and an n squared has to appear in the answer.

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