Sequences, Explicit & Recursive

A sequence is a function whose inputs are the counting numbers. Describe it from the start, or from the term before.

A pattern is built from tiles. Stage 1 has 4, stage 2 has 7, stage 3 has 10.

How many tiles at stage 100? You could draw ninety-seven more pictures.

a growing pattern, stage by stage stage 1 4 tilesstage 2 7 tilesstage 3 10 tilesstage 4 13 tiles each stage adds the same three tiles — one to every row

Or look at what each stage actually does. Every stage adds one tile to each of the three rows — three more, every time.

So the count goes 4, 7, 10, 13, …, always stepping by 3.

Stage 100 has taken 99 steps of 3 from the start: 4 + 99 × 3 = 301.

Notice the 99, not 100. The first stage needed no steps at all — which is exactly why the formula reads a1 + (n-1)d, and why that -1 causes more errors here than anything else.

A growing tile pattern with rule 3n plus 2 n = 15n = 28n = 311n = 414 step of 3 each time, plus a starting 2 → rule: 3n + 2

Arithmetic or geometric?

Subtract consecutive terms. Constant? Arithmetic, with that common difference d.

Divide consecutive terms. Constant? Geometric, with that common ratio r.

Neither? It may still have a pattern — squares, or Fibonacci — but it is not one of these two families.

Test both ways

Always try subtracting and dividing. Two quick checks settle which family you are in, and everything after that follows from the formula.

Explicit formulas

Arithmetic: an = a1 + (n - 1)d. The (n-1) is there because the first term needs no steps.

Geometric: an = a1 · rn-1, for the same reason.

Explicit formulas let you jump straight to the 100th term without computing the other 99.

Off by one

The 10th term uses 9 steps, not 10. Writing a1 + nd instead of a1 + (n-1)d is the most frequent error in this whole unit.

Recursive formulas

A recursive definition needs two lines: the starting value and the rule.

Arithmetic: a1 = 4, an = an-1 + 3. Geometric: a1 = 4, an = 2an-1.

Without the starting value the rule describes infinitely many different sequences, so it is not optional.

Both lines, always

A recursive answer that omits a1 is incomplete and loses marks, even when the rule is perfect.

Sequences as functions

The domain of a sequence is the counting numbers 1, 2, 3, … — not all real numbers.

So its graph is a set of separate dots, never a continuous line. Do not join them up.

An arithmetic sequence sits on a straight line; a geometric one sits on an exponential curve.

Worked examples

Which family?

Classify 80, 40, 20, 10, … and write an explicit formula.

  1. Differences: -40, -20, -10 — not constant. Not arithmetic.
  2. Ratios: 4080 = 12, and the same each time. Geometric with r = 12.
  3. an = 80 (12)n-1 First term times the ratio, n-1 times.
  4. Check a4 = 80(18) = 10 ✓ Matches the list.

Geometric; an = 80(12)n-1

Explicit to a far term

An arithmetic sequence has a1 = -5 and d = 6. Find a25.

  1. an = -5 + (n - 1)(6) The arithmetic formula.
  2. a25 = -5 + 24(6) 24 steps to reach the 25th term.
  3. = -5 + 144 = 139 Equivalently 6n - 11, and 6(25) - 11 = 139 ✓

a25 = 139

Writing both forms

Write explicit and recursive formulas for 3, 12, 48, 192, ….

  1. Ratios: 123 = 4, constant. Geometric with r = 4.
  2. Explicit: an = 3 · 4n-1 Check a3 = 3(16) = 48 ✓
  3. Recursive: a1 = 3 and an = 4an-1 Starting value plus the rule — both lines.

an = 3 · 4n-1; or a1 = 3, an = 4an-1

Words to know

Sequence
An ordered list of numbers. The nth one is an.
Explicit formula
A rule giving an directly from n.
Recursive formula
A rule giving an from an-1, plus a starting value.
Common difference
The constant added each time, in an arithmetic sequence.
Common ratio
The constant multiplied each time, in a geometric sequence.

Mistakes to watch for

Using n where the formula needs n - 1. The tenth term is a1 + 9d, because the first term is already there before any step is taken. Count the gaps, not the terms.

Giving a recursive rule without the first term. "an = an-1 + 3" describes infinitely many sequences — one for every possible start. Both lines are the answer, not just the interesting one.

Joining the dots when graphing a sequence. Its domain is the counting numbers, so there is no stage 2.5. The graph is separate points, and a connecting line claims something that does not exist.

Testing only for a common difference. If subtracting gives nothing constant, try dividing. 80, 40, 20, 10 has no common difference at all but a perfectly clean ratio of 12.

Asked to keep a class busy, a teacher told the young Carl Friedrich Gauss to add up every number from 1 to 100.

He is said to have answered almost immediately: 5050. He had paired 1 with 100, 2 with 99, and so on — fifty pairs each summing to 101.

He was about nine. The trick works for any arithmetic sequence, and it is still the standard proof of the sum formula.

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