Finding the nth Term of a Quadratic Sequence

Say you are laying a mosaic border round a pond, ring by ring, each ring one tile further out. Ring one takes two tiles, ring two takes nine, ring three takes twenty. Ring four takes thirty-five and ring five takes fifty-four. The tile shop wants one number before it cuts anything: the count for ring twelve. The jumps between those counts run seven, then eleven, then fifteen, then nineteen. Not one of them repeats, so there is no add-the-same-amount shortcut hiding in this border. You could grind up to ring twelve by hand, but ring fifty would defeat you, and one early slip poisons every ring after it. You want an expression that jumps straight to any ring you name.

Before any algebra happens, there is a check that tells you what shape of answer you are hunting for. Write the counts in a row and take the difference between each pair of neighbours. Seven, eleven, fifteen, nineteen. Those first differences refuse to settle, so this sequence is not linear. Now take the differences of those differences. Eleven take away seven is four. Fifteen take away eleven is four. Nineteen take away fifteen is four, every single time. A second difference that is constant and not zero means the sequence is quadratic. Its rule has to carry an n squared, and any answer without one is wrong before you check a number in it. Every quadratic rule has the same three-part shape, written in letters as a n squared, plus b n, plus c on the end. If that second row keeps moving instead, the video called Finding the nth Term of a Cubic Sequence takes that case, and spotting quadratic growth from a picture of tiles is the video called Recognising Special Sequences. So why does a second difference decide anything. Take the plain square numbers, one, four, nine, sixteen, twenty-five, whose rule is nothing but n squared. Their first differences grow, three, five, seven, nine, but their second differences are two, two, two. A pure n squared sequence always sits on a second difference of two. Double every term and every difference doubles too, so the second difference is always twice the number sitting in front of n squared. Quick check, and you have everything you need. A quadratic sequence has a second difference of six. Does its rule start with six n squared, three n squared, or twelve n squared? Pick one. I'll wait. The answer is three n squared. The second difference comes out at twice that coefficient, so you halve it, and six halved is three. Back on the border, the second difference is four, so the rule starts with two n squared. That came out of two rows of subtraction and no algebra at all.

Three stages turn that opening piece into the whole expression, and they always run in the same order. Your handle for this video is three words: halve it, subtract it, finish it. Halve the second difference, subtract the squared part away, then finish whatever is left standing. Stage one is already behind you. Four halved is two, so two n squared is riding somewhere inside this rule. Stage two is the one people try to do in their heads. Underneath the sequence, build a second row: put n equals one, two, three, four and five into two n squared. That row reads two, eight, eighteen, thirty-two, fifty. It is exactly what the squared part contributes to each ring. Now subtract it from the original, one column at a time. Two take away two leaves nothing. Nine take away eight leaves one. Twenty leaves two, thirty-five leaves three, fifty-four leaves four. What survives is zero, one, two, three, four. That is a brand new sequence, with the squared part lifted clean out of it. Stage three treats those leftovers as an ordinary linear sequence. They rise by one, so the n part is one n. At position one, n gives one while the leftover is zero, so take one away: n minus one. That is the method from the video called Finding the nth Term of a Linear Sequence, run on leftovers instead of originals. Push the pieces together and you have it. Ring n takes two n squared, plus n minus one, tiles. Test it on a middle term before you trust it. Put n equals four in. Two times sixteen is thirty-two, add four for thirty-six, take one, thirty-five. Ring four exactly. Your turn, and this is the number the tile shop was waiting for. Use that expression to find how many tiles ring twelve needs. Have a go at this one. I'll wait. Twelve squared is one hundred and forty-four, doubled is two hundred and eighty-eight. Add twelve for three hundred, take one away, and ring twelve needs two hundred and ninety-nine tiles. One expression, one substitution, and no climbing through the rings in between. Ring fifty would cost those same three lines.

This method also settles a formula you were probably handed and told to take on faith. The triangular numbers go one, three, six, ten, fifteen. Their second differences are one, one, one. One halved is a half, so the rule opens with a half n squared, and a fraction in front is perfectly allowed. Take that row away and the leftovers are a half, one, one and a half, two, two and a half. They climb by a half from a half, so they are a half n, and the rule tidies into half of n, times n plus one. That is the triangular formula the video called Recognising Special Sequences asks you to take as given, with nothing behind it but a second difference of one.

A paper can tell you outright that a sequence is quadratic, and that word changes what your answer is allowed to look like. Here is a sequence a question describes as quadratic: five, fourteen, twenty-seven, forty-four, sixty-five. A student answers nine n minus four, taking the nine from the first gap, five up to fourteen. Test it and you see why it survives a careless look. Position one gives five, correct. Position two gives fourteen, correct again. Position three gives twenty-three, and the sequence says twenty-seven. You never needed that arithmetic. Nine n minus four carries no n squared, so it cannot be quadratic, and it goes in the bin on sight. Here is one line from an examiner report, about a question where students were told the sequence in front of them was quadratic. Despite being told that it was a quadratic sequence most students gave an expression for the nth term of a linear sequence. They usually put six in front of n, with six n plus four and six n plus one popular. Most students scored one mark for working out the next two terms, although there were several arithmetic errors made in that calculation They were handed the word quadratic and still wrote a straight line. Reading that word and putting an n squared on the page are two separate acts. The other thing reports flag is not the coefficient at all. This line comes from a different report, on a question about quadratic sequences. In question fifteen, on quadratic sequences, many candidates found the value of a but were unable to progress to b and c because they did not know how to subtract the values of three a squared. They had the coefficient and stalled on stage two, the subtraction, because they attempted it in their heads. Write the squared row underneath, subtract down the columns, and there is nothing left to hold. So take that sequence back: five, fourteen, twenty-seven, forty-four, sixty-five. Run all three stages and find the expression the student should have written. Take your time. I'll wait right here. Differences first. The first differences are nine, thirteen, seventeen, twenty-one, and the second differences are four, four, four. Halve that four for two n squared. The two n squared row is two, eight, eighteen, thirty-two, fifty. Subtract it and the leftovers are three, six, nine, twelve, fifteen, climbing in threes from three, so they are three n. The correct expression is two n squared plus three n, and nine n minus four never stood a chance, because it had no squared part to strip out.

Halve it, subtract it, finish it. That is the short version, and here is the whole thing in one place. Take differences twice. A second difference that is constant and not zero means quadratic, so your answer has to carry an n squared. Halve that second difference for the number in front of n squared, because a pure n squared sequence sits on a second difference of two. Write the squared row underneath the sequence and subtract it away column by column, on paper, never in your head. Treat what is left as a fresh linear sequence, find its rule the ordinary way, then add the two pieces into one expression. And check the finished expression on a middle term, because a wrong answer can still match the first two terms perfectly.

Next in the chapter: Finding the nth Term of a Cubic Sequence, where the second difference keeps moving and you take the differences one row further down.

For more, visit scholafly.com, or watch the next video.