Arithmetic Progressions: nth Term, Sum and Word Problems
Arithmetic Progressions carries a short question on the common difference and a 3–4 mark question on a term, a sum or a word problem.
Core formulas
For first term a and common difference d: nth term aₙ = a + (n − 1)d; sum of the first n terms Sₙ = n/2[2a + (n − 1)d], or Sₙ = n/2(a + l) when the last term l is known. Also aₙ = Sₙ − Sₙ₋₁, and the number of terms n = (l − a)/d + 1.
Handy devices: sum of the first n natural numbers is n(n + 1)/2; three terms in AP can be taken as a − d, a, a + d; four as a − 3d, a − d, a + d, a + 3d.
Worked example
The 5th term of an AP is 19 and the 10th is 39. Find a, d and S₂₀. a + 4d = 19 and a + 9d = 39. Subtracting: 5d = 20, so d = 4; then a = 19 − 16 = 3. S₂₀ = 20/2[2(3) + 19(4)] = 10(6 + 76) = 820.
How questions are framed
- "Which term of the AP is 78?" - set a + (n − 1)d = 78 and solve for the integer n.
- "How many terms?" - use n = (l − a)/d + 1 first, then sum.
- "Sum of multiples of 3 between 100 and 500" - identify a, d and l before touching a formula.
Traps that cost marks
- Using nd instead of (n − 1)d shifts every term by one step.
- With negative d the bracket arithmetic goes wrong; write the bracket fully before multiplying.
- If Sₙ is given and aₙ is wanted, use Sₙ − Sₙ₋₁, not the aₙ formula.
- n must come out a positive integer; a fraction or negative signals a slip, so recheck a and d.