An Arithmetic Progression (AP) is a sequence in which each term differs from the previous by a fixed number called the common difference (d). General form: a, a+d, a+2d, … Example: 2, 5, 8, 11 (d = 3).

The nth term of an AP: aₙ = a + (n–1)d, where a = first term, d = common difference, n = term number. Example: For AP 3, 7, 11…, a₁₀ = 3 + 9×4 = 39.

Sₙ = n/2 × [2a + (n–1)d] or Sₙ = n/2 × (a + l), where l is the last term. Example: Sum of first 20 terms of 1, 3, 5… = 20/2 × [2×1 + 19×2] = 10 × 40 = 400.

APs are used in real-life problems: calculating savings over time, finding number of seats in rows, determining total production over days. If Sₙ is given, aₙ = Sₙ – Sₙ₋₁. For three terms in AP use: a–d, a, a+d; for four terms use: a–3d, a–d, a+d, a+3d.

Arithmetic Progressions appear in many real-life contexts: savings growing by a fixed amount each month, seats in a stadium arranged in rows with a fixed increase, salary increments, stacking of logs in a triangular pattern. Example: A person saves ₹100 in January, ₹120 in February, ₹140 in March... This is an AP with a=100, d=20. Total savings in 12 months = S₁₂ = 12/2 × [2×100 + 11×20] = 6 × 420 = ₹2520. Recognising AP patterns helps solve real problems quickly.