3.2PureCore

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence. Two equivalent formulas give the sum of the first n terms instantly — no need to add them one by one.

25 min Sequences and Series

Edexcel A level Maths: 3.2 Arithmetic SeriesCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Use the sum formula Sₙ = n/2(2a + (n−1)d)
Use the alternative Sₙ = n/2(a + l)
Find n given a target sum
Apply the sum of natural numbers

The sum formulas

The sum of the first $n$ terms has two equivalent forms — use the first when you know $a$ and $d$ , the second when you know the first term $a$ and last term $l$ .

S_n = \tfrac{n}{2}\big(2a + (n-1)d\big)

Using first term and common difference.

S_n = \tfrac{n}{2}(a + l)

Using first and last terms.

Example 1Find the sum of the first 20 terms of 3, 7, 11, …

a = 3

d = 4

n = 20

S_{20} = \tfrac{20}{2}(2(3) + 19(4)) = 10(6 + 76)

Answer

820

Finding n

If a target sum is given, substitute into $S_n$ to get a $\textbf{quadratic in } n$ , then solve (discard non-positive or non-integer solutions).

Tip — Setting Sₙ equal to a value gives a quadratic in n — solve and keep the sensible (positive integer) root.

Sum of natural numbers

A special case: $1 + 2 + \cdots + n = \tfrac{n(n+1)}{2}$ (here $a = 1$ , $d = 1$ ). This appears often in proofs and sigma-notation work.

1 + 2 + \cdots + n = \tfrac{n(n+1)}{2}

Sum of the first

n

natural numbers.

Formula recap

S_n = \tfrac{n}{2}(2a + (n-1)d)

Sum (a and d).

S_n = \tfrac{n}{2}(a + l)

Sum (first and last).

\sum_{1}^{n} r = \tfrac{n(n+1)}{2}

Sum of 1…n.

Common mistakes to avoid

Using the geometric sum formula for an arithmetic series.

Arithmetic series use n/2(2a + (n−1)d), not a(1−rⁿ)/(1−r).

Keeping a negative or fractional n.

n must be a positive integer — discard other roots.

Key takeaways

Sₙ = n/2(2a + (n−1)d) or n/2(a + l).
A target sum gives a quadratic in n.
1 + 2 + … + n = n(n+1)/2.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

The sum of an arithmetic series is:
- $\tfrac{n}{2}(2a + (n-1)d)$
- $a + (n-1)d$
- $\tfrac{a(1-r^n)}{1-r}$
- $ar^{n-1}$
Show answer and explanation
Answer: $\tfrac{n}{2}(2a + (n-1)d)$
The arithmetic sum formula.
The sum of the first 20 terms of 3, 7, 11, … is:
- $820$
- $800$
- $420$
- $1640$
Show answer and explanation
Answer: $820$
10(6 + 76) = 820.
$S_n = \tfrac{n}{2}(a + l)$ uses the first term and the:
- Last term
- Common difference
- Common ratio
- Number of terms only
Show answer and explanation
Answer: Last term
l is the last term.
$1 + 2 + \cdots + 100$ equals:
- $5050$
- $10000$
- $5000$
- $255$
Show answer and explanation
Answer: $5050$
100(101)/2.
Setting Sₙ equal to a given value produces:
- A quadratic in n
- A linear equation
- A geometric series
- No equation
Show answer and explanation
Answer: A quadratic in n
n appears squared.
When solving for n, you keep values that are:
- Positive integers
- Any real number
- Negative
- Fractions
Show answer and explanation
Answer: Positive integers
n counts terms.

Next step

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Related lessons

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.