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 Video by Zeeshan Zamurred Sequences and Series

Edexcel A level Maths: 3.2 Arithmetic SeriesWatch the full walkthrough before the notes below.

Open on YouTube

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

Using first term and common difference.

S_{n} = \frac{n}{2} (a + l)

Using first and last terms.

a = 3

d = 4

n = 20

S_{20} = \frac{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 $quadratic in n$ , 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: $1 + 2 + \dots + n = \frac{n ( n + 1 )}{2}$ (here $a = 1$ , $d = 1$ ). This appears often in proofs and sigma-notation work.

1 + 2 + \dots + n = \frac{n ( n + 1 )}{2}

Sum of the first

n

natural numbers.

Formula recap

S_{n} = \frac{n}{2} (2 a + (n - 1) d)

Sum (a and d).

S_{n} = \frac{n}{2} (a + l)

Sum (first and last).

1 \sum n r = \frac{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

Ready to lock in Arithmetic Series? Pick a mode and earn XP & Dobloons.