Simple Arithmetic Progression (AP Series)

Arithmetic Progression (AP) series are among the most common number-series patterns in aptitude tests. They are important because they teach you to recognise steady, linear change - a skill useful for spotting simple trends quickly during exams.

The key idea: Each term in an Arithmetic Progression (AP) increases or decreases by a fixed constant value - called the common difference (d).

In such a series, if the first term is a₁ and the common difference is d, then every term can be obtained by repeatedly adding (or subtracting) d.

General Form of an AP:
a1, a1 + d, a1 + 2d, a1 + 3d, …

Formulas to Remember:
• n-th term: an = a1 + (n - 1) × d
• Common difference: d = a2 - a1
• Sum of first n terms: Sn = n/2 × [2a1 + (n - 1)d]
• (Alternate form) Sn = n/2 × (a1 + an)

Key Notes:
• If d > 0, the series is increasing.
• If d < 0, the series is decreasing.
• If d = 0, all terms are equal (constant series).
• Every linear pattern where the difference remains the same is an AP.

1. Decreasing AP: terms reduce by a constant (e.g., 50, 45, 40, 35 → d = -5).

2. AP with non-integer d: common difference may be fractional (e.g., 1.5, 2.25, 3.0 → d = 0.75).

3. AP hidden inside longer patterns: AP might appear on every 2nd or 3rd term (use sub-series extraction).

• Always compute consecutive differences first (quick scan of two or three gaps).
• If differences are constant, use Next = Last + d; if not, check for alternating sub-series or higher differences.

• Identify the common difference (d) by subtracting two consecutive terms.
• If d is constant, the series is an AP and next term = last term + d (or use a_n = a₁ + (n - 1)d).
• For decreasing AP, d will be negative - apply the same add rule (which subtracts numerically).
• If differences are not constant, split into sub-series or check higher-order differences before concluding.

Example to remember:
Series: 3, 7, 11, 15 → d = 4 → Next = 19