An AP plots as points on a straight line

0:00Here is the tile pattern again, growing by four at every stage. One tile, then five, then nine, then thirteen, then seventeen. Write it as two rows. The stage number along the top, the tile count underneath. Stage one, one. Stage two, five. Stage three, nine. Stage four, thirteen. Stage five, seventeen. And a last column for stage n, which holds four n minus three. Two rows like this are already halfway to a picture.

0:34Because a table with a position above a value is a list of pairs, and pairs are what you plot. Take one column at a time and write it as a pair. One and one. Two and five. Three and nine. Four and thirteen. Five and seventeen. But before any of those go on a grid, one decision has to be made and stated. Which of the two numbers runs across, and which runs up?

1:03The convention here is position across, value up. Stage number on the horizontal, tile count on the vertical. That is not a free choice you can quietly reverse later. Swap them and the steepness you read off turns into one over the common difference, and every reading in the picture changes. So: position first, value second, and it stays that way. Now put them on. One across, one up. Two across, five up. Three across, nine up.

1:39Four across, thirteen up. Five across, seventeen up. And the thing your eye does before you tell it to is join them. They line up. All five of them, on one straight line. That is worth pausing on, because the obvious reaction is that this example happened to work out nicely. It did not happen to. It could not have done anything else. Look at what moving from one dot to the next actually involves.

2:11One step to the right, because the positions go up by one. Four steps up, because the values go up by four. One right, four up. Then one right, four up again. Then again, and again. Lay a staircase over the dots and every tread is the same width and every riser is the same height. Equal steps across with equal rises is not evidence of a straight line. It is what a straight line is.

2:41The common difference being constant and the rise being constant are the same sentence twice. So a progression whose gap never changes cannot produce a bend. There is nothing for the bend to come from. The algebra says it too, once you push the brackets out. The n th term is a plus, n minus one, times d. Multiply that out and you get a plus d n minus d.

3:11Collect the parts that do not move: d n plus, a minus d. That is a rule of degree one in n. The position appears once and never squared. For the tiles, a is one and d is four, so d n plus a minus d is four n minus three. Degree one is exactly the shape whose graph is a straight line. So the picture was decided before anything was plotted.

3:40Now read the picture backwards. Put a right-angled step anywhere on the line: one across, and however far up it takes you. It takes you up four. The steepness is four. And four is the common difference. The steepness of the plot is d itself. Which means you can be handed a plot with no formula attached and recover the rule from it. Better than that, the positions you happened to mark do not matter.

4:10Plot only stages three, five and seven, at nine, seventeen and twenty-five, and the line through them is the same line. Read it back at position one and it hands you one, the first term, from a picture where position one was never marked. Follow the line the other way, left and down, until it meets the vertical axis. It crosses at minus three. The tempting thing to say is that the crossing is the first term. It is not.

4:44The first term is one. The crossing is minus three. In the general form the crossing is a minus d. One common difference before the sequence starts. The line arrives at the axis one step early, because the axis is at position zero and the sequence begins at position one. The only progressions where the crossing really is the first term are the ones that never move at all. There, a minus d is a, because d is nothing.

5:18Which brings us to the line itself, and a warning the picture does not give you. The line is a reading aid. The sequence is the dots. Ask the line what it holds at position two and a half and it answers seven, quite happily. There is no stage two and a half. There is no term there. Nothing lives at that point. And it is not only the halfway places.

5:44Between minus five and zero the line hands back a whole number at every single position, six of them, all of them sitting neatly on the ruling. Every one of those is a place the sequence refuses. So the graph of a sequence is a set of separated points, and the line joining them was drawn by us, for us. Two more, and this time predict before plotting. Two, five, eight, eleven. The gap is three, so it is a progression, and its rule is three n minus one.

6:21Minus five, minus one, three, seven. The gap is four, so it is one too, and its rule is four n minus nine. Before a single dot goes down you can say what both plots will look like. Both straight. The first rising three for every step across, the second rising four. The second is the steeper of the two. And check the crossings against the rule: minus one for the first, minus nine for the second.

6:53In each case the first term minus one common difference, exactly as promised. Steeper does not mean bigger, and that is worth seeing on a grid. Here is one progression starting at fifty and climbing by one. And here is another starting at two and climbing by five. The second is five times as steep and, for a long while, far behind. At position five, the gentle one is at fifty-four and the steep one is at twenty-two.

7:26They meet at position thirteen, where both are sixty-two. Before that the gentle one leads at every position. After it, the steep one does, and never gives the lead back. Steepness reports the step. It says nothing about where a progression started. A bend tells you something too. One, four, nine, sixteen, twenty-five. Those gaps are three, five, seven, nine. Not constant, so not a progression of this kind, and the plot curves away.

8:01But notice what the bend does not say. That sequence has a perfectly good rule: n times n. A bend rules out a constant step. It does not rule out a rule. Three, six, twelve, twenty-four, forty-eight bends far harder, and it has a rule too. So the shape of a plot is a diagnosis. Straight says constant step. Bent says the step is changing, and nothing more. One last piece of care, and it cuts the other way.

8:35If a plot bends, the sequence is certainly not one of these. That direction is safe. But five dots lying on a line do not prove the sequence is. Here is a sequence whose first five values are one, five, nine, thirteen and seventeen. Plotted, it is the same five dots on the same line, indistinguishable from the tile pattern. Its sixth value is a hundred and forty-one. The line says twenty-one.

9:07The dots agreed everywhere anyone looked, and disagreed at the first place nobody did. So the picture is a strong hint and a fast way to read the step off. The proof is still the gaps.