Arithmetic Progressions and Series
Arithmetic Progressions and Series examines whether you can identify a common difference, find a named term, and add a stated number of terms. The formulas are simple; the marking traps are in counting.
Core rules
- An AP has a first term a and common difference d = t₂ − t₁; the nth term is tₙ = a + (n − 1)d.
- Sum of the first n terms: Sₙ = n/2[2a + (n − 1)d] = n/2(a + l), where l is the last term.
- The sum of the first n natural numbers is n(n + 1)/2.
- If a, b, c are in AP, then 2b = a + c.
- To find n when the last term is given, solve a + (n − 1)d = l and check that n is a positive integer.
Worked example
For the AP 3, 7, 11, …, find the 10th term and the sum of the first 10 terms. Here a = 3 and d = 4. t₁₀ = 3 + 9×4 = 39. S₁₀ = 10/2(3 + 39) = 5×42 = 210. The same sum using Sₙ = n/2[2a + (n − 1)d] gives 5[6 + 36] = 210.
Traps
The nth term uses (n − 1)d, not nd. For “sum of the first 20 terms” do not accidentally use n = 19. A decreasing AP has negative d, so subtract rather than add. If a question says “terms between 10 and 100 divisible by 7”, the first term is 14 and the last is 98 - listing the endpoints wrongly shifts every count.