Arithmetic Progressions — Cheat Sheet

Class X · Mathematics Ch.5 · 8 Panels · Quick Reference

🔢 What is an AP?
a, a+d, a+2d, a+3d, …
each term = previous term + d
a = first termThe starting term a₁
d = common differenceFixed number added each step
aₙ = nth termThe general term
Definition: A list of numbers where the difference between any two consecutive terms is constant.
➕ Common Difference (d)
d = aₙ − aₙ₋₁
d = a₂ − a₁ = a₃ − a₂ = …
Value of dType of APExample
d > 0Increasing2, 5, 8, …
d < 0Decreasing10, 7, 4, …
d = 0Constant5, 5, 5, …
Test: A list is an AP only if aₖ₊₁ − aₖ is the SAME for every k.
📍 nth Term Formula
aₙ = a + (n − 1)d
nth from end = l − (n − 1)d
What each symbol means
afirst term
dcommon difference
nterm number
llast term = aₙ
Example: 3, 7, 11, … → a₁₀ = 3 + 9×4 = 39.
Is x a term? Set aₙ = x and solve for n. If n is a positive whole number → yes; else no.
➗ Sum of First n Terms
Sₙ = n/2 [2a + (n − 1)d]
Sₙ = n/2 (a + l)  (when l known)
aₙ = Sₙ − Sₙ₋₁
Second form: Sum = (no. of terms) × (average of first & last term).
Gauss trick: 1 + 2 + … + 100 = 100/2 × (1 + 100) = 5050.
Note: Sum of first n natural numbers = n(n+1)/2.
🎯 Smart Term Selections
Terms in APChoose as
3 termsa−d, a, a+d
4 termsa−3d, a−d, a+d, a+3d
5 termsa−2d, a−d, a, a+d, a+2d
Why? These symmetric choices make the d-terms cancel when you add, so a sum condition instantly gives a.
✨ Key Properties
  • Add/subtract a constant to each term → still an AP (same d)
  • Multiply each term by k → AP with common difference kd
  • If aₙ is linear in n (like 3n+2) → the list is an AP
  • If Sₙ is quadratic in n (like 2n²+n) → the list is an AP
Middle term: In an AP, each term is the average of its neighbours: aₖ = (aₖ₋₁ + aₖ₊₁)/2.
🌍 Common Applications
MoneySalary increments, fixed savings, instalments
StackingLogs, bricks, seats in rows
NumbersMultiples in a range, natural number sums
Approach: Identify a and d from the situation, decide which formula (aₙ or Sₙ) fits the question, then solve.
🧠 Exam Quick Recall
aₙ = a + (n−1)d
Sₙ = n/2 [2a + (n−1)d]
Sₙ = n/2 (a + l)
d = a₂ − a₁  |  aₙ = Sₙ − Sₙ₋₁
  • Always verify n comes out as a positive whole number
  • Reject impossible values (negative counts, fractions of items)
  • For "which term", solve aₙ = value for n
  • For "how many terms sum to S", solve Sₙ = S for n
Careful: Sₙ formula uses (n−1)d, NOT nd. A very common slip!