Arithmetic Sequences: nth Term and Sum Formulas + Examples

An arithmetic sequence is one where the difference between consecutive terms is always the same. That constant difference is called the common difference, written b. For example 3, 7, 11, 15, 19 has a common difference of 4. This follows directly from number patterns in Year 8 and is developed at senior level.

Sequence or Series?

These two words are often treated as one, yet they differ — and that accounts for half the errors in this topic.

• Sequence — the list itself: 3, 7, 11, 15
• Series — the sum of it: 3 + 7 + 11 + 15 = 36

If the question asks for "the 10th term", that is a sequence. If it asks for "the sum of the first 10 terms", that is a series. Reading the keyword first saves more marks than learning an extra formula.

The nth Term Formula

where a is the first term, b the common difference and n the term number.

Find b by subtracting any term from the one after it: b = U₂ − U₁.

Example: For 5, 9, 13, 17, ... what is the 25th term?

a = 5, b = 4 → U₂₅ = 5 + (25−1) × 4 = 5 + 96 = 101

The Sum of the First n Terms

There are two forms, and they are equivalent:

Use the first when the last term is known, the second when it is not.

Example: The sum of the first 20 terms of 3, 7, 11, ...

a = 3, b = 4, n = 20

S₂₀ = 20/2 × (2×3 + 19×4) = 10 × (6 + 76) = 10 × 82 = 820

Where the Sum Formula Comes From

The classic story of the young Gauss, told to add the numbers 1 to 100. He paired 1+100, 2+99, 3+98 — each pair totalling 101, with 50 pairs. Answer: 5,050, in seconds.

That is exactly what Sₙ = n/2 × (a + Uₙ) says: the number of pairs times the value of each pair. A child who has seen this derivation never wonders whether to multiply by n or n/2.

The Middle Term

When the number of terms is odd there is a middle term:

Negative Common Differences

The sequence 20, 17, 14, 11 is also arithmetic, with b = −3. The formulas are identical — there is no special rule for decreasing sequences, a question that comes up often and has a reassuring answer.

A Worked Word Problem

A trader saves Rp 50,000 in the first month and adds Rp 10,000 more each month than the month before. What is the total saved after 12 months?

a = 50,000, b = 10,000, n = 12

S₁₂ = 12/2 × (2×50,000 + 11×10,000) = 6 × (100,000 + 110,000) = 6 × 210,000 = Rp 1,260,000

The word "total" marks this as a series question, not a sequence one.

The Mistakes That Actually Cost Marks

1. Using n instead of (n−1). The 25th term uses 24, not 25. The answer is then out by exactly one common difference — if a child is consistently over or under by exactly b, this is why.

2. Answering with a term when a sum was asked for. A reading error, not a maths one.

3. Computing the difference backwards. The common difference is the later term minus the earlier one. For 20, 17, 14 it is −3, not 3.

4. Confusing it with a geometric sequence. Arithmetic adds a fixed difference; geometric multiplies by a fixed ratio. Quick test: subtract two consecutive terms. Constant result means arithmetic.

5. Forgetting to divide by 2 in the sum formula. The answer comes out exactly double — an instantly recognisable error pattern.