Introduction
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.
Pattern: Simple Arithmetic Progression (AP Series)
Pattern
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.
Step-by-Step Example
Question
Find the next term in the series: 2, 5, 8, 11, 14, ?
Solution
-
Step 1: Compute consecutive differences
5 - 2 = 3, 8 - 5 = 3, 11 - 8 = 3, 14 - 11 = 3. -
Step 2: Identify the common difference
Since all consecutive differences = 3, common difference d = 3. This confirms an AP. -
Step 3: Apply the n-th term rule or add d to last term
Next term = Last term + d = 14 + 3 = 17. -
Final Answer:
17 -
Quick Check:
Verify difference: (17 - 14) = 3 ✅ - differences remain constant, series is AP.
Quick Variations
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).
Trick to Always Use
- 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.
Summary
Summary
- 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 an = 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
Hint: Check for constant addition or subtraction between terms.
Common Mistakes: Adding a different number assuming pattern changes.