An arithmetic series is the sum of the terms of an arithmetic sequence. The sum of the first n terms is Sₙ = n/2[2a + (n − 1)d], or Sₙ = n/2(a + l) when the last term l is known. Setting Sₙ equal to a total gives a quadratic in n, sigma notation is an arithmetic series in disguise, and a sum condition combined with a term condition gives simultaneous equations for a and d.
Arithmetic series
- Sₙ = n/2[2a + (n − 1)d]
- Sₙ = n/2(a + l), where l is the last term
- Σ (from r = 1 to n) of (pr + q): first term p + q, common difference p, n terms
- Sum of terms k + 1 to n: Sₙ − Sₖ
The two sum formulae
Pairing the first term with the last, the second with the second-last and so on gives n/2 pairs, each adding to a + l. That is Sₙ = n/2(a + l). Replacing l with a + (n − 1)d gives the other form, Sₙ = n/2[2a + (n − 1)d]. Use whichever matches the information you have, and remember that n is the number of terms, not the value of the last term.
Finding n from a sum
Setting Sₙ equal to a total gives a quadratic in n. Multiply through by 2 to clear the fraction, rearrange to zero and solve, keeping the positive whole-number root. “The first time the total exceeds” a value works the same way as an inequality: find the boundary, then round up and check.
Sigma notation
Σ (r = 1 to n) of (pr + q) is an arithmetic series. Substitute r = 1 for the first term and r = n for the last term, then use Sₙ = n/2(a + l). Watch the limits: a sum from r = 5 to r = n has n − 4 terms, not n.
Sum and term conditions together
A sum condition and a term condition give two equations in a and d. Expand the sum fully, for example S₈ = 8a + 28d, before pairing it with the term equation, then solve simultaneously. The sum of a block of terms, such as the 21st to the 30th, is S₃₀ − S₂₀.
Totals in context
Cumulative amounts such as savings, repayments or production that increase by a fixed amount each period form arithmetic series. Identify a as the first period's amount, d as the change per period and n as the number of periods, then use Sₙ. Asking how long until a total is reached gives a quadratic in n, or can be checked by evaluating neighbouring sums.
Worked examples
Example 1. Find the sum of the first 20 terms of 3, 7, 11, …
- a = 3, d = 4, n = 20.
- S₂₀ = 10[6 + 19 × 4] = 10 × 82.
Answer: 820.
Marks: M1 for the substitution, A1 for 820.
Example 2. How many terms of 2, 5, 8, … are needed for a sum of 155?
- n/2[4 + 3(n − 1)] = 155, so n/2(3n + 1) = 155.
- 3n² + n − 310 = 0, which factorises as (3n + 31)(n − 10) = 0.
- Take the positive whole-number root.
Answer: 10 terms.
Marks: M1 for the quadratic, M1 for solving, A1 for n = 10.
Example 3. Evaluate Σ (r = 1 to 20) of (3r + 2).
- First term (r = 1): 5. Last term (r = 20): 62. Number of terms: 20.
- S = 20/2 × (5 + 62) = 10 × 67.
Answer: 670.
Marks: M1 for the sum formula with a and l, A1 for 670.
Example 4. Given S₈ = 100 and u₃ = 8, find a and d.
- S₈ = 4[2a + 7d] = 100, so 2a + 7d = 25.
- u₃ = a + 2d = 8, so a = 8 − 2d.
- 2(8 − 2d) + 7d = 25, so 3d = 9 and d = 3, giving a = 2.
Answer: a = 2, d = 3.
Marks: M1 for 2a + 7d = 25, M1 for solving, A1 for both.
Example 5. Ravi saves £10 in week 1, then £3 more each week than the week before. Find his total after 12 weeks, and the week in which his total first exceeds £500.
- S₁₂ = 6[20 + 11 × 3] = 6 × 53 = 318.
- Sₙ = n/2(3n + 17). S₁₅ = 465 and S₁₆ = 520.
Answer: £318 after 12 weeks; the total first exceeds £500 in week 16.
Marks: M1 for S₁₂, M1 for testing totals, A1 for week 16.
Common mistakes
- Confusing n with the last term. n is the number of terms; find it from the last term using uₙ = a + (n − 1)d.
- Using n instead of n − 1. The formula is n/2[2a + (n − 1)d].
- Keeping a negative root for n. Only a positive whole number of terms makes sense.
- Miscounting terms in sigma notation. From r = 5 to r = 20 there are 16 terms, not 20.
- Not expanding the sum before solving. Write S₈ fully in a and d before pairing it with another equation.
How arithmetic series are examined
Arithmetic series questions are set on WMA12, the Pure 2 paper in Edexcel International A-Level; 9MA0 students study them in their second year. Questions ask you to:
- find the sum of a number of terms, or of a block of terms
- find n given a total, by solving a quadratic
- evaluate a sum written in sigma notation
- model savings or repayments, and find when a total is reached
These question types, with full mark schemes, appear among the sequences and series exam questions for Pure 2. Use the Pure 2 sequences and series worksheet or the sequences and series revision engine for more.
This builds on arithmetic sequences. Compare the methods with geometric series.
Frequently asked questions
What is the sum formula for an arithmetic series?
Sₙ = n/2[2a + (n − 1)d], or Sₙ = n/2(a + l) if you know the last term l.
How do you find the sum of the 21st to the 30th terms?
Work out S₃₀ − S₂₀.
How do you evaluate a sigma sum like Σ(3r + 2)?
Find the first and last terms by substituting the limits, count the terms, then use Sₙ = n/2(a + l).