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Level 10Grade 10skill: sequences· 10 min read

Number Sequences: Finding the Rule

In short: Don't answer with the next term — find the rule. Constant first differences mean a linear rule dn + c, where c is the 'zeroth term' (first term − d). Constant second differences mean a quadratic an² + bn + c, where a is HALF the second difference. Keep the minus sign on decreasing sequences, and always verify at n = 1 and n = 2.

How to find the rule of a sequence

A sequence question almost never wants the next term. It wants the rule — a formula for the nth term that works for term 1, term 100, and term 5,000 alike. The next term is one number; the rule is the whole pattern. Read the question twice before you compute anything.

Start with differences. Subtract each term from the one after it. If those first differences are constant, the sequence is linear (arithmetic) and the rule looks like dn + c. If the first differences keep changing but their own differences settle to a constant, the sequence is quadratic and the rule looks like an² + bn + c.

The single move that saves you: for a linear sequence, step one place backwards to the 'zeroth term'. If the sequence is 7, 11, 15, 19 with common difference 4, the term before 7 is 7 − 4 = 3. That zeroth term is exactly the c in 4n + c, so the rule is 4n + 3 with no guessing.

  1. Write the first differences between consecutive terms.
  2. If they are constant (= d), the sequence is linear: build the dn part, then add the zeroth term — the value one step before the first term, which equals first term − d.
  3. If they are not constant, write the second differences. If those are constant (= D), the sequence is quadratic and the coefficient of n² is D ÷ 2 — half the second difference, never the whole thing.
  4. For a quadratic, subtract (D/2)n² from every term; what remains is a linear sequence you solve the same way to get b and c.
  5. State the FULL nth-term rule, then substitute n = 1 and n = 4 to check it reproduces the sequence.
  6. Only now answer what was asked — the rule itself, a specific far-out term, or which term hits a target value.

Why the second difference is halved Why it works

Linear vs quadratic
Constant FIRST differences → linear (dn + c). Constant SECOND differences → quadratic (an² + bn + c).

6, 10, 14, 18 has first differences 4, 4, 4 → linear. 6, 10, 16, 24 has first differences 4, 6, 8 and second differences 2, 2 → quadratic.

The zeroth-term trick
For a linear sequence, the constant c equals the term you'd get by stepping one place BEFORE term 1. It saves you from solving simultaneous equations.

30, 25, 20, 15 has d = −5. Step back: 30 − (−5) = 35. Rule is −5n + 35. Check n = 1: −5 + 35 = 30.

Half the second difference
If the nth term is an² + bn + c, the second difference is always 2a. So a = (second difference) ÷ 2 — the most-missed step in the whole topic.

5, 12, 23, 38, 57 has second difference 4, so a = 2 (NOT 4). The rule is 2n² + n + 2, and 2 × 1² + 1 + 2 = 5.

Here is the insight worth keeping: differencing is just discrete calculus. First differences behave like a first derivative and second differences like a second derivative. A straight line has constant slope, so a linear sequence has constant first differences; a parabola an² + bn + c has a constant second derivative equal to 2a, so a quadratic sequence has constant second differences equal to 2a. That is exactly why you divide by two — you are undoing the '2' that differentiating n² twice leaves behind.

Worked examples

Find the nth-term rule for 7, 11, 15, 19, … and then the 20th term. = Rule: 4n + 3; the 20th term is 83.
  1. First differences: 11 − 7 = 4, 15 − 11 = 4, 19 − 15 = 4. Constant, so linear with d = 4.
  2. Zeroth term = 7 − 4 = 3, so the rule is 4n + 3.
  3. Check: n = 1 gives 4 + 3 = 7, and n = 4 gives 16 + 3 = 19. Correct.
  4. 20th term: 4 × 20 + 3 = 83.
A sequence is 20, 17, 14, 11, … . Find the rule, then find which term equals −7. = Rule: −3n + 23; the 10th term equals −7.
  1. First differences: −3, −3, −3. Linear and decreasing, so d = −3 (keep the minus sign).
  2. Zeroth term = 20 − (−3) = 23, so the rule is −3n + 23.
  3. Set −3n + 23 = −7. Then −3n = −30, so n = 10.
  4. Check: −3 × 10 + 23 = −30 + 23 = −7. Correct.
Find the nth-term rule for the quadratic sequence 3, 8, 15, 24, 35, … = Rule: n² + 2n (equivalently n(n + 2)).
  1. First differences: 5, 7, 9, 11 — not constant, so try again.
  2. Second differences: 2, 2, 2. Constant, so quadratic with coefficient of n² equal to 2 ÷ 2 = 1.
  3. Subtract n² from each term: 3−1, 8−4, 15−9, 24−16, 35−25 = 2, 4, 6, 8, 10. That leftover is linear with rule 2n.
  4. Combine: n² + 2n. Check n = 4: 16 + 8 = 24. Correct.
Find the nth-term rule for 5, 12, 23, 38, 57, … — a strong-student trap. = Rule: 2n² + n + 2 (the n² coefficient is 2, not 4).
  1. First differences: 7, 11, 15, 19. Second differences: 4, 4, 4.
  2. The tempting error is to make the n² coefficient 4. But the coefficient is HALF the second difference: 4 ÷ 2 = 2.
  3. Subtract 2n² from each term: 5−2, 12−8, 23−18, 38−32, 57−50 = 3, 4, 5, 6, 7. That is linear with rule n + 2.
  4. Combine: 2n² + n + 2. Check n = 3: 18 + 3 + 2 = 23, and n = 5: 50 + 5 + 2 = 57. Correct.

Common mistakes

Answering with the next term instead of the rule

✗ 23

For 7, 11, 15, 19 the sequence keeps rising by 4, so the next term is 19 + 4 = 23. That is a real number in the sequence — but the question asked for the nth-term RULE, not the fifth term. A single number can never describe every position.

Fix: Underline what is asked. If it says 'nth term', 'rule', or 'formula', your answer must contain an n. Here that is 4n + 3, and 23 is merely the value at n = 5.

wrong_operation

Building dn but forgetting the zeroth term

✗ 4n

You find the common difference 4 and jump straight to 4n. But 4n gives 4, 8, 12, 16 — every value is 3 too small, because you never added the constant. You built the slope and dropped the intercept.

Fix: After the dn part, always add the zeroth term: (first term − d). Here 7 − 4 = 3, so the rule is 4n + 3, not 4n. Test n = 1 immediately; if it doesn't give the first term, your constant is missing.

order_of_ops_lr

Stopping at the common difference

✗ 4

You correctly compute that consecutive terms differ by 4 and write 4 as the answer. But 4 is the common difference, not the rule. It tells you how the sequence grows, not how to find term n.

Fix: The common difference is step one of three, not the finish line. Turn it into dn, then add the zeroth term: 4 → 4n → 4n + 3. The rule always contains n.

partial_computation

Dropping the minus on a decreasing sequence

✗ 3n + 17

For 20, 17, 14, 11 you notice the terms are '3 apart' and write d = 3, then 20 − 3 = 17, giving 3n + 17. It even passes the n = 1 check (3 + 17 = 20) — the trap. But n = 2 gives 23, not 17. A falling sequence cannot have a positive slope.

Fix: For a decreasing sequence the common difference is negative: d = −3. Then the zeroth term is 20 − (−3) = 23 and the rule is −3n + 23. Always test a SECOND term (n = 2), because the first term alone can hide a sign slip.

sign_error

Practice

Work through a mix of linear and quadratic sequences. Each asks for the nth-term rule, a far-out term, or which position hits a target — so decide what is being asked before you compute. Watch the traps you just met: give the rule (not the next term), add the zeroth term, and keep the minus sign on decreasing sequences.

Related skills

First, make sure of

Frequently asked questions

How do I know whether a sequence is linear or quadratic?

Take the first differences between consecutive terms. If they are constant, the sequence is linear and the rule is dn + c. If the first differences keep changing, take the differences of those (the second differences); if THOSE are constant, the sequence is quadratic with rule an² + bn + c.

Why is the coefficient of n² half the second difference, not the whole thing?

Differencing an² + bn + c twice always leaves 2a. So the constant second difference equals 2a, which means a is the second difference divided by 2. If the second difference is 4, the n² coefficient is 2, not 4 — this is the most common trap in the topic.

What is the difference between the next term and the nth-term rule?

The next term is a single number you get by continuing the pattern once. The nth-term rule is a formula containing n that gives ANY term instantly. If a question says 'rule', 'formula', or 'nth term', your answer must contain an n — a bare number cannot be the rule.

How do I avoid sign errors on a decreasing sequence?

Keep the common difference negative from the start: for 20, 17, 14, 11 use d = −3, not 3. Find the zeroth term as first − d = 20 − (−3) = 23, giving −3n + 23. Always test n = 2 as well as n = 1, because the first term alone can pass even when the sign is wrong.

What is the fastest way to find the constant term c in a linear rule?

Step one place backwards to the 'zeroth term' — the value at n = 0. It equals the first term minus the common difference, and it IS the constant c. For 7, 11, 15 with d = 4, the zeroth term is 7 − 4 = 3, so the rule is 4n + 3 with no simultaneous equations needed.