Find arithmetic progression terms and sums
Use the nth-term and finite-sum formulas to solve arithmetic progression problems, including contexts.
Section 12 · Series
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the nth-term formula for an AP with first term a and difference d?
What you will learn
- Find the nth term of an arithmetic progression
- Find the sum of its first n terms
- Apply the formulas to a stated finite context
The nth term advances by n − 1 equal differences
For first term a and common difference d, uₙ = a + (n − 1)d. The first term uses zero jumps; the nth term uses n − 1 jumps.
To find an unknown position, substitute the term value and solve for n. Check that a position number is a positive integer.
Pair the first and last terms to find a finite sum
The sum of the first n terms is Sₙ = n/2 [2a + (n − 1)d], equivalently n/2(a + uₙ). These formulas are given in the syllabus formula list.
Use the first-and-last form when uₙ is already known. Keep negative common differences and units consistent in contextual questions.
Pause and explain
For an AP with a = 5 and d = 3, what is u₄?
Worked example
Find the 20th term and the sum of the first 20 terms of 7, 11, 15, 19, …
Show the worked solution
- Here a = 7, d = 4 and n = 20.
- u₂₀ = 7 + 19 × 4 = 83.
- S₂₀ = 20/2(7 + 83) = 10 × 90 = 900.
Answer 20th term 83; sum of the first 20 terms 900.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
For the AP 5, 8, 11, … find the 12th term and the sum of its first 12 terms.
Give me a hint
Use a = 5, d = 3 and pair the first and last terms for the sum.
Compare my reasoning
- u₁₂ = 5 + 11 × 3 = 38.
- S₁₂ = 12/2(5 + 38).
- Calculate 6 × 43.
12th term 38; sum 258.
Look for these in your work
- Term formula applied
- Correct endpoints used
- Sum evaluated
For 23, 19, 15, … find the 15th term and sum the first 15 terms.
Give me a hint
The common difference is negative; keep its sign in both formulas.
Compare my reasoning
- a = 23 and d = −4.
- u₁₅ = 23 + 14(−4) = −33.
- S₁₅ = 15/2(23 − 33) = −75.
15th term −33; sum −75.
Look for these in your work
- Negative difference retained
- Nth term checked
- Finite sum sign and value checked
A theatre has 18 seats in the first row, and each row has 2 more seats than the one before. How many seats are there across the first 15 rows?
Give me a hint
Treat the row counts as a finite AP and sum them.
Compare my reasoning
- a = 18, d = 2 and n = 15.
- The 15th row has 18 + 14 × 2 = 46 seats.
- Total = 15/2(18 + 46) = 480 seats.
480 seats.
Look for these in your work
- Context mapped to a and d
- Last row calculated
- Total includes all 15 rows
Common mistakes
- Using n differences instead of n − 1
- Using the nth term as the common difference
- Adding terms one by one when a sum formula is quicker
- Dropping units from a contextual total
What can you explain now?
In an AP, how many common-difference jumps separate the first term from the 12th term?
Compare with the explanation
11
The nth-term formula uses n − 1 differences.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How do uₙ and Sₙ use the same first term and difference in an arithmetic progression?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.