Unit 24: Patterns in a Row

Arithmetic sequences

3, 7, 11, 15, 19 … goes up by 4 every time. A sequence with a constant step is called arithmetic.

To jump straight to any term, notice that the 5th term has taken four steps, not five:

term = first + (n - 1) × step

So the 10th term here is 3 + 9 × 4 = 39.

Geometric sequences multiply

2, 6, 18, 54 … does not add each time — it multiplies by 3. That is a geometric sequence.

To jump ahead, multiply that many times:

term = first × ratio^(n - 1)

Geometric sequences grow terrifyingly fast, which is why they turn up in stories about grains of rice on chessboards.

Add them from both ends

The story goes that a teacher told young Gauss to add up 1 to 100, expecting an hour of peace. He answered in seconds.

He paired the ends: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101. Fifty pairs, each worth 101.

50 × 101 = 5050.

The general rule

The trick works for any arithmetic sequence: every pair from the ends adds to the same amount.

So the sum is the average of the first and last, times how many there are:

sum = n × (first + last) / 2

It is not a formula to memorise. It is Gauss’s pairing, written down.

A formula for every term at once

The sequence 5, 8, 11, 14 goes up by 3, so its rule involves 3n.

But 3n gives 3, 6, 9, 12 — always 2 short. So the rule is 3n + 2.

Step first, then adjust. Check it on the first term: 3 × 1 + 2 = 5. Correct.

Is a number in the sequence?

Is 47 in the sequence 3n + 2?

Set 3n + 2 = 47, so 3n = 45 and n = 15. A whole number, so yes — it is the 15th term.

If n came out as a fraction, the answer would be no. That is the whole test, and it is unit 3 again.

Square and triangular numbers

The square numbers are 1, 4, 9, 16, 25 — dots arranged in a square.

The triangular numbers are 1, 3, 6, 10, 15 — dots stacked in a triangle. Each one adds the next whole number, so the nth is 1 + 2 + … + n, which by unit 24 lesson 2 is n(n + 1)/2.

Gauss’s pairing gives them instantly.

Two triangles make a square

Add two consecutive triangular numbers: 3 + 6 = 9. 6 + 10 = 16. 10 + 15 = 25.

Always a square number. Two triangles of dots fit together into a square, which you can check by drawing.

So n(n+1)/2 + (n+1)(n+2)/2 simplifies to (n+1)^2 — a fact you could discover by playing and then prove with unit 7.

Fibonacci

1, 1, 2, 3, 5, 8, 13, 21 — each term is the sum of the two before it.

There is no simple nth-term formula you would want to use; the rule refers back to the sequence itself.

It turns up in pine cones, sunflowers and rabbit populations, and the ratio of one term to the next creeps towards about 1.618 — the golden ratio.

Finding the ratio

For a geometric sequence, divide any term by the one before it. That is the ratio, and it is the same all the way along.

For 3, 12, 48: 12/3 = 4 and 48/12 = 4. The ratio is 4.

If those two divisions disagree, the sequence is not geometric, and that check is worth doing first.

The missing middle

If 4 and 36 are two terms with one missing between them, what is it?

The ratio must be the same both times, so x/4 = 36/x, giving x^2 = 144 and x = 12.

That is the geometric mean — the square root of the product, rather than the average. It sits between them in a multiplying sense.

Three sums that come up constantly

The whole numbers to n: n(n + 1)/2.

The first n odd numbers: exactly n^2. So 1 + 3 + 5 + 7 = 16.

The first n powers of 2, starting at 1: 2^n - 1. So 1 + 2 + 4 + 8 = 15.

Each one has a picture behind it, and each saves a page of adding.

Adding a stretch in the middle

To add the numbers from 20 to 30, do not start again from scratch.

Add everything to 30, then subtract everything to 19: 465 - 190 = 275.

That "whole take away the part you did not want" move is the same one from the shaded-area lessons, and it works just as well on sums.

A geometric series has its own trick

To add 2 + 6 + 18 + 54, call the total S. Then 3S = 6 + 18 + 54 + 162.

Subtract the first from the second and almost everything cancels: 2S = 162 - 2, so S = 80.

Same idea as Gauss — find a way to write it that makes the middle disappear.

When it never ends but still stops

Add 1/2 + 1/4 + 1/8 + 1/16 … forever. Each step goes half the remaining distance to 1, so the total creeps towards 1 and never passes it.

When the ratio is between -1 and 1, an endless geometric series has a finite total:

sum = first / (1 - ratio)

Which is why 0.999… is 1

0.999… is really 9/10 + 9/100 + 9/1000 … — an endless geometric series with ratio 1/10.

Its total is (9/10)/(1 - 1/10) = 1.

So 0.999… = 1 exactly. Not nearly. The same number, written twice.