# The Mathematical Ninja and The $n$th Term

*Note: this post is only about arithmetic and quadratic sequences for GCSE. Geometric and other series, you’re on your own.*

Quite *how* the Mathematical Ninja had set up his classroom so that a boulder would roll through it at precisely that moment, the student didn’t have time to ponder. He ran along in front of it, with the Indiana Jones theme playing in his head for a few seconds before realising he could simply sidestep and allow the boulder to crash through the wall.

“I guess it’s not $n+4$, then,” said the student.

The Mathematical Ninja smiled a thin smile. “You’re right. The $n$th term of the arithmetic sequence that starts 9, 13, 17… is not $n+4$. That would give you 5, 6, 7, 8…”

“Ah,” said the student. “But my dad said…”

“I will set up a boulder for your dad in due course. Instead, you need to think about a sequence you know goes up in fours.”

“How about… the four times table?”

Nod. “Only we call it $4n$.”

“I see… the sequence $4n$ starts 4, 8, 12… but it’s five short? So we need to… add 5! Is it $4n+5$?”

“There you go.”

### How about quadratic sequences?

“I’ve got one here that goes -1, 1, 11, 29…”

The Mathematical Ninja raised an eyebrow. “That’s a tricky one. What do you notice?”

“Well, the differences keep on changing - it’s down 6, up 2, up 10, up 18… so the differences *in the differences* go up by 8 each time.”

“Good!” said the Mathematical Ninja. “Can you think of another sequence whose second difference goes up by 8 each time?”

“No,” said the student, without really thinking, and then caught himself. “Well - I know that the square numbers start 1, 4, 9, 16…, and their differences are 3, 5, 7…, which go up by *two* each time.”

“So, if $n^2$ has a second difference that goes up by 2, what has a second difference that goes up by 8?”

“Um… maybe $4n^2$? So that starts 4, 16, 36, 64… (Hey, it’s the even squares! How about that?). Anyhow, should we see how far away that is from the sequence?”

“Try it!”

“Five too high, then 15 too high, then 25 too high… that’s going up in tens. Oh - but I know how to do those - it’s too high by $10n - 5$!”

“Keep going!”

“So I need to take that away, to give me $4n^2 -10n$… $+ 5$?”

A thumb up. “Better check it, though!”

“If $n=4$, I get $4\times 16 - 10 \times 4 + 5 = 64 - 40 + 5 = 29$. It works!”