🪜Arithmetic & Geometric Sequences

Glaciologist Ren drills a core out of Slope Glacier every spring and writes the depth on the wall of the hut. After five years the wall reads

12, 19, 26, 33, 40

That is a sequence: an ordered list of numbers. The order is the whole point. Each number has a position, and the notation puts the position in a subscript: a_1 is the first term, a_2 the second, and a_n is the term at position n — read aloud as "the nth term".

Here a_1 = 12, a_3 = 26 and a_5 = 40. Notice what a_n is *not*: it is not n, and it is not a multiple of n. The position and the term are two different numbers, and telling them apart is the single most useful habit in this topic. When a question asks "which row has 40 seats?" it is asking for n. When it asks "how many seats are in row 12?" it is asking for a_12.

Because every position gets exactly one term, a sequence is a function — a function whose input is a whole number, 1, 2, 3, ..., instead of any number at all. That is why the rest of this lesson looks so much like the work on lines and on growth curves. It is the same mathematics, sampled at the whole numbers.

There are two families worth naming, and you can tell them apart with two quick tests.

• ◆Subtract each term from the one after it. If every difference is the same, the sequence is arithmetic, and that shared value is the common difference d.
• ◆Divide each term by the one before it. If every quotient is the same, the sequence is geometric, and that shared value is the common ratio r.

Run the tests in that order, and run them on *every* pair, not just the first. In 12, 19, 26, 33 the differences are 7, 7, 7 — arithmetic, with d = 7. In 3, 6, 12, 24 the differences are 3, 6, 12, which are not equal, but the ratios are 2, 2, 2 — geometric, with r = 2. A sequence like 1, 4, 9, 16 passes neither test: the differences 3, 5, 7 grow and the ratios 4, 2.25, 1.78 shrink, so it is neither.

An arithmetic sequence adds the same amount at every step, so its terms rise (or fall) in a straight line. If you plotted position across and term up, the points would sit on a ruler.

To get from a_1 to a_n you take steps of size d. How many? Not n. One fewer than n, because the first term is already there before any step is taken. From a_1 to a_4 is three steps, not four. That gives the formula for the nth term:

a_n = a_1 + (n - 1)d

For 12, 19, 26, 33, ... with a_1 = 12 and d = 7:

a_20 = 12 + (20 - 1) x 7 = 12 + 19 x 7 = 12 + 133 = 145

and no term in between had to be written down. That is the point of a formula: it jumps straight to the position you want.

Two details cause almost every wrong answer here.

• ◆The multiplier is n - 1, not n. If in doubt, test your formula at n = 1: it has to hand a_1 straight back.
• ◆A falling sequence has a negative d. In 45, 41.5, 38, 34.5, ... the difference is 41.5 - 45 = -3.5, not 3.5. The size of the gap is only half the answer; the direction is the other half.

Compare the formula with y = mx + b and the family resemblance is obvious. Multiplying out gives a_n = dn + (a_1 - d), so d is the slope and a_1 - d is the intercept — the height the line would have at position 0, one step before the sequence starts.

A geometric sequence multiplies by the same amount at every step. Where an arithmetic sequence climbs a staircase, a geometric one scales.

The step count works exactly as before — from a_1 to a_n is n - 1 steps — except that each step is a multiplication, and repeated multiplication is a power:

a_n = a_1 * r^(n - 1)

For 3, 6, 12, 24, ... with a_1 = 3 and r = 2:

a_7 = 3 * 2^(7 - 1) = 3 * 2⁶ = 3 x 64 = 192

The exponent, not the coefficient, is where the ratio lives. Writing 3 x 2 x 6 instead of 3 x 2⁶ is the same mistake as adding when you should multiply, and it is worth catching early by asking whether the answer grew *fast enough*.

The common ratio does not have to be a whole number, and it does not have to be positive.

• ◆r between 0 and 1 makes the sequence shrink. 48, 24, 12, 6, ... has r = 1/2, and a snow-globe that rebounds to 3/4 of the height it fell from has r = 3/4.
• ◆A negative r makes the signs alternate: 2, -6, 18, -54, ... has r = -3. Odd powers of a negative number stay negative, even powers turn positive, so the sign of a term depends on whether n - 1 is odd or even.

One thing to be careful about: r is a term divided by the term before it, so r = a_2 / a_1. Taking a_1 / a_2 instead gives the reciprocal, and 2 turns into 1/2 — a sequence that shrinks instead of growing.

Given two terms that are not neighbours you can still find r, because the gap tells you how many multiplications happened. If a_2 = 2 and a_5 = 54 then three multiplications separate them, so r³ = 54 / 2 = 27 and r = 3. The same reasoning finds d in an arithmetic sequence: if a_3 = 11 and a_8 = 31 then five steps carried the value up by 20, so d = 20 / 5 = 4, and from there a_1 = 11 - 2 x 4 = 3.

There are two honest ways to describe a sequence, and they answer different questions.

An explicit formula gives a_n directly in terms of n. "a_n = 3n + 2" answers "what is the 40th term?" in one line.

A recursive rule gives a term in terms of the one before it, plus a starting value. It is written in two parts, and both parts are needed:

a_1 = 5, a_n = a_(n - 1) + 3, for n > 1

Read that as "start at 5, and each new term is the previous term plus 3". It describes exactly how the sequence is built, which makes it the natural language for anything that grows step by step — savings, populations, the height of each bounce. Its weakness is that to reach a_40 you must walk all thirty-nine steps.

A rule with no starting value is not a rule at all: "a_n = a_(n - 1) + 3" describes 5, 8, 11, ... and 100, 103, 106, ... equally well. That is why every recursive rule names a_1.

Converting recursive to explicit is a matter of reading off a_1 and the step.

• ◆a_1 = 5, a_n = a_(n - 1) + 3 is arithmetic with d = 3, so a_n = 5 + (n - 1) x 3 = 3n + 2.
• ◆a_1 = 4, a_n = 3 * a_(n - 1) is geometric with r = 3, so a_n = 4 * 3^(n - 1).

Converting explicit to recursive goes the other way, and the trap is the starting value.

• ◆For a_n = 5n - 2, the coefficient of n is the common difference, so each step adds 5. The first term is not -2: substitute n = 1 to get a_1 = 5 - 2 = 3. The rule is a_1 = 3, a_n = a_(n - 1) + 5.
• ◆For a_n = 3 * 2^(n - 1), the base of the power is the ratio, so each step multiplies by 2, and a_1 = 3 * 2⁰ = 3. The rule is a_1 = 3, a_n = 2 * a_(n - 1).

Not every recursive rule makes an arithmetic or a geometric sequence. "a_1 = 2, a_n = 2 * a_(n - 1) + 3" gives 2, 7, 17, 37, ... which is neither, and the only way to reach a_4 is to take the steps one at a time.

The answer box is plain text, so here is exactly what each kind of question wants.

• ◆A term, a common difference or a position is a whole number: type 145, or -6, or 12. If the prompt says "which row", it wants the row number, not the number of seats.
• ◆A common ratio or a term that is not whole is a fraction in lowest terms: type 1/2, 3/4, 16/9. Write 3/4, never 6/8, and never 0.75 for a question that asked for a fraction.
• ◆A decimal answer comes with its rounding written into the prompt ("to 1 decimal place"), and you should give exactly that: -56.5.
• ◆An explicit formula is typed as an expression in n, right-hand side only. Type 3n + 2, not "a_n = 3n + 2" — the checker reads the formula, and the subscript notation is not something it can parse. Write multiplication with * and powers with ^, so a geometric formula looks like 5*2^(n - 1). Keep the bracket around n - 1: without it, 5*2ⁿ - 1 means something else entirely.
• ◆A recursive rule and "arithmetic, geometric or neither" come as options to choose from, because a two-part rule cannot be typed into one box. Read both halves of each option — the starting value and the step — since the wrong ones usually get one half right.

Any equivalent form of a formula is accepted, so 5 + (n - 1) x 3 passes just as 3n + 2 does. Before you submit one, test it at n = 1: if it does not give back the first term of the list you were shown, it is describing a different sequence.