Finding the nth Term of a Cubic Sequence

A three by three puzzle cube sits on a lot of desks. Count it as a solid block and there are three small cubes along every edge, which makes twenty-seven altogether. The bigger versions carry on the same way. Four along an edge counts as sixty-four small cubes, five along an edge as one hundred and twenty-five. Watch what that does to the jumps. Three to four adds thirty-seven pieces. Four to five adds sixty-one. Growing in three directions at once makes the increases grow too. Now a real problem. A display is built in stages, and the number of small cubes runs three, ten, twenty-nine, sixty-six, one hundred and twenty-seven. Someone asks you for stage twenty. The gaps between those terms are seven, nineteen, thirty-seven and sixty-one, so there is no common difference. The gaps between the gaps are twelve, eighteen and twenty-four, so they have not settled either. That looks like the differences method breaking down on you. It has not broken down at all, it simply has not finished yet.

Every cubic sequence carries the same fingerprint, and the plain cube numbers show it most clearly. Write out one, eight, twenty-seven, sixty-four and one hundred and twenty-five. That is n cubed, for the positions one to five. The first differences are seven, nineteen, thirty-seven and sixty-one. The second differences are twelve, eighteen and twenty-four. Neither row has stopped changing. So take one more row. Eighteen take twelve is six, and twenty-four take eighteen is six as well, so the third differences are six and six. It has settled. A cubic sequence always settles on the third row, and here is the reason. Each round of subtracting knocks the power down by one. Cubed drops to squared, squared drops to a straight line, a straight line drops to a constant. Three rounds, three rows. For n cubed on its own, the number that row settles on is six, every single time. Six is the cube's fingerprint. Now multiply every term by three, giving three, twenty-four, eighty-one, one hundred and ninety-two. Every difference triples along with it, because subtracting one tripled number from another gives a tripled answer. That third row now lands on eighteen, which is three sixes. So the third difference is always six times the number sitting in front of n cubed. Here is a quick one for you. A sequence has a constant third difference of eighteen. Is the number in front of n cubed eighteen, six, or three? Take your pick. I'll wait. The answer is three, because eighteen is six threes. You divide the third difference by six to get the coefficient, and copying eighteen straight in would build a sequence six times too big. The third difference divided by six is where every answer in this topic begins.

Back to those display numbers, and this time the table gets built all the way down. Stages one to five gave three, ten, twenty-nine, sixty-six and one hundred and twenty-seven. Write them along a row with space underneath. Subtract each term from the one after it, and the first differences are seven, nineteen, thirty-seven and sixty-one. Do the same to that row. Nineteen take seven is twelve, thirty-seven take nineteen is eighteen, sixty-one take thirty-seven is twenty-four. So the second differences are twelve, eighteen and twenty-four, and they are still climbing. Do you stop there and call this a quadratic sequence, or do you take one more row? Have a think. I'll wait. You take one more row. A second difference that is still changing is the signal to keep going, never the signal to force an n squared answer onto the sequence. Eighteen take twelve is six, and twenty-four take eighteen is six. The third differences are constant, so this sequence is cubic. Divide that six by six and the number in front of n cubed is one, so the answer starts with a plain n cubed. Now take the n cubed part away from every term. Under three write one, under ten write eight, under twenty-nine write twenty-seven, and so on. Three take one is two. Ten take eight is two. Twenty-nine take twenty-seven is two, and the last two positions leave two as well. What is left over is a flat two in every position, so the nth term is n cubed plus two. Write that as an expression in n, not as a sentence about what you add on. Test it on a term you can see. Stage four: four cubed is sixty-four, add two and you get sixty-six, which is exactly what the sequence gave. Stage twenty is now a five second job instead of an afternoon of counting. Twenty cubed is eight thousand, add two, so eight thousand and two small cubes.

The second example changes one thing, and it is the thing most people get wrong. A sequence begins one, fifteen, fifty-three, one hundred and twenty-seven, two hundred and forty-nine, and the question wants an expression for the nth term. Build the difference table down to a constant row, then use that row to get the number in front of n cubed. That first stage is all I want from you here. Pause it there and work it out. I'll wait. The first differences are fourteen, thirty-eight, seventy-four and one hundred and twenty-two. The second differences are twenty-four, thirty-six and forty-eight. The third differences are twelve and twelve. Twelve divided by six is two, so this answer starts with two n cubed. The coefficient is not always one. The cube numbers you meet as a pattern are simply the case where it happens to be one. Two n cubed for the first five positions gives two, sixteen, fifty-four, one hundred and twenty-eight and two hundred and fifty. Take those away from the sequence and every position leaves minus one. One take two is minus one, fifteen take sixteen is minus one, and it holds all the way along. So the nth term of that sequence is two n cubed minus one. Both examples left a flat number behind, which is what a simple cubic sequence does. If your leftovers are still changing, that leftover row is a sequence of its own, and Finding the nth Term of a Quadratic Sequence finishes it off. Two signals are worth keeping. If the second differences are already constant, stop on row two, because that sequence is quadratic rather than cubic. And if no row ever settles however far you go, differences are the wrong tool, and Finding the nth Term of an Exponential Sequence handles that family.

The wording of the question decides what shape your answer has to take. Take a question like this one: find an expression for the nth term. An expression means a formula written in n, one you can substitute any position into. Two n cubed minus one answers that. A description of the jumps, such as add fourteen, then thirty-eight, then seventy-four, does not, because it never hands you a formula to put a position into. The other answer that fails is twelve n cubed minus one, from copying the third difference straight into the front of the formula. Put n equals one into that and it gives eleven, when the sequence starts at one. One line of checking stops that. Substitute n equals one into your finished expression and compare it against the first term the question printed. Leave the difference rows on the page rather than rubbing them out, and write the final line as an expression in n, or as T n equals that expression if the question uses that letter.

Six is the cube's fingerprint. Here is the whole method again, in five lines, and then you are free to go. One: take differences, and keep taking them until a row stops changing. A cubic sequence settles on the third row, because three rounds of subtracting take a cubed term down to a flat number. Two: divide that constant third difference by six, and what you get is the number in front of n cubed. Three: subtract that n cubed part away from every term of the original sequence. Four: whatever is left over, add it on to give the full expression. Five: substitute a position in and check your answer against a term you can see. And when the second differences keep on changing, that is not the method failing you. It is the method telling you there is one more row to take.

Next in the chapter: Finding the nth Term of an Exponential Sequence, where the differences never settle at any level and a power takes over instead.

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