📊 Study Notes – Tables
(Quantitative Aptitude – IIT‑JEE & Competitive Exams)
Why Tables?
Tables form the backbone of many aptitude problems: you’re asked to fill missing entries, deduce a rule, or find a hidden pattern. Mastering tables means you can solve a wide variety of problems quickly and accurately.
1. Overview
| Sub‑type | Typical Pattern | Example |
|---|---|---|
| 1.1 | Arithmetic Progression (AP) | 2, 5, 8, 11, … |
| 1.2 | Geometric Progression (GP) | 3, 9, 27, … |
| 1.3 | Quadratic / Polynomial | 1, 4, 9, 16, … (n²) |
| 1.4 | Fibonacci / Recursive | 1, 1, 2, 3, 5, 8, … |
| 1.5 | Factorial / Powers | 1, 2, 6, 24, … (n!) |
| 1.6 | Multiplication / Division Tables | 1×1, 1×2, … |
| 1.7 | 2‑D Tables (Row × Column patterns) | See section 2 |
| 1.8 | Miscellaneous (Triangular, Catalan, etc.) | 1, 1, 2, 5, 14, … |
Tip: Before computing, look for symmetry or invariants (e.g., sum of row = sum of column, product of corners = center²).
2. General Strategy for Solving Tables
- Identify the dimension (rows × columns).
- Check for a single pattern that applies to all rows or all columns.
- Test the pattern on a subset (2–3 rows/cols) to confirm.
- Cross‑validate: the pattern should satisfy every entry, not just the visible ones.
- Use algebraic representation:
- Let
R_ibe the rule for row i,C_jfor column j. - Often the entry at (i, j) =
f(i, j)(e.g.,i + j,i × j,i² + j², …).
- Let
- If multiple patterns are plausible, seek the one that gives integer or simple values for all missing entries.
- Check constraints:
- Only positive integers?
- Are values ≤ some bound?
- Are the entries within a certain range or obey divisibility?
3. Common Patterns & Their Algebraic Forms
| Pattern | Formula | Key Properties |
|---|---|---|
| 3.1 Arithmetic Progression (AP) | a_n = a_1 + (n-1)d |
Constant first difference d. |
| 3.2 Geometric Progression (GP) | a_n = a_1 r^(n-1) |
Constant ratio r. |
| 3.3 Quadratic (n²) | a_n = n² |
Second difference constant = 2. |
| 3.4 Cubic (n³) | a_n = n³ |
Third difference constant = 6. |
| 3.5 Fibonacci | F_n = F_(n-1)+F_(n-2) |
Recursive; requires two seeds. |
| 3.6 Factorial | n! = 1×2×…×n |
Exponential growth. |
| 3.7 Multiplication Table | a_{i,j} = i × j |
Symmetric: a_{i,j}=a_{j,i}. |
| 3.8 Division Table | a_{i,j} = i ÷ j |
Often leads to fractions; test for integer |