1. AS Maths Masterclass Lesson 1: Arithmetic series

2. Learning objectives The student should be able to: recognise an Arithmetic Progression (AP); recall the formula for the sum to n terms; evaluate the terms and sum of a given AP; manipulate formulae that model APs.

3. What do the following have in common ? 5 + 7 + 9 + 11 + 13 + … … + 29 – 8 – 5 – 2 + 1 + 4 40 + 30 + 20 + 10 + 0 – 10 – 20 – 30 – 40

4. “They all have a difference (d) in common!” E.g. Take 5 + 7 + 9 + 11 + … + 29 Each term is bigger than its previous term by 2 So Also In general Or

5. “Let’s go straight to the nth term” We have that And that And further that In general: Click here for weblink 2 Click here for spreadsheet

6. Proof of the sum to n terms If we write out the terms of the series we get If we now write out these terms in reverse order Adding each pair of terms we then get And so

7. Finding a formula for First take the sum formula: Then substitute a = 1, d = 1 to get So 1+2+3+…+100 = 50 x 101 = 5050 etc

8. Arithmetic Progression Example The 5 th term of an AP is and the 7 th term of the same AP is Find a and d. Well, writing down the nth terms (n = 5,7) gives Subtracting gives from which Substituting this in either equation leads to