Skip to main content
MathidiaMathematics Academy
Sequences and binomial expansion
Lesson 16 of 75
Introduction

Arithmetic sequences and series

Objective. Use nth-term and sum formulae with contextual constraints.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

An arithmetic sequence increases or decreases by a constant common difference d between consecutive terms, with nth term formula aₙ=a+(n−1)d, where a is the first term. The sum of the first n terms is Sₙ=n/2(2a+(n−1)d), useful when a context gives a starting value and constant rate of change, such as weekly savings increasing by a fixed amount. For a=5 and d=3, the 10th term is 5+9×3=32, and the sum of the first 10 terms is 10/2(10+27)=185. Always identify a and d correctly from the context before substituting into either formula.

Worked example

Find the sum of the first 20 terms of the arithmetic sequence with first term 4 and common difference 5.

20 … 4
Illustrated card showing 20 … 4
  1. 1Identify a=4, d=5, n=20
  2. 2Use Sₙ = n/2(2a+(n−1)d)
  3. 3Substitute: S₂₀ = 10(8+19×5) = 10(8+95)
  4. 4Simplify to get S₂₀ = 1030

Answer: 1030

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. Students use n instead of (n−1) in the nth term formula, computing the 5th term of a=2, d=4 as 2+5×4=22 instead of the correct 2+4×4=18; the common difference is only added (n−1) times.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard