Study Notes – Series in Reasoning Ability
Target Exam: IIT‑JEE / Competitive Entrance Exams
Section: Reasoning Ability – Series
Focus: Rigorous derivation, exam‑oriented strategies, and a hard practice problem.
1. What is a Series?
A series is a sequence of numbers (or symbols) arranged in a particular order according to a rule (or pattern). In reasoning tests, the goal is to:
- Identify the rule/pattern.
- Predict the next term(s), fill a missing term, compute a sum, etc.
2. Common Types of Number Series
| Type | General Rule | Sample Pattern | Key Features |
|---|---|---|---|
| Arithmetic (AP) | (a_n = a_1 + (n-1)d) | 2, 5, 8, 11, … | Constant difference (d). |
| Geometric (GP) | (a_n = a_1 \cdot r^{,n-1}) | 3, 6, 12, 24, … | Constant ratio (r). |
| Harmonic | (a_n = \frac{1}{b_n}) where (b_n) is AP | 1, ½, ⅓, ¼, … | Reciprocal of AP. |
| Fibonacci | (a_n = a_{n-1}+a_{n-2}) | 1, 1, 2, 3, 5, 8, … | Sum of two preceding terms. |
| Quadratic / Polynomial | (a_n = an^2 + bn + c) | 1, 4, 9, 16, … | Second difference constant. |
| Mixed / Hybrid | Combines two/three rules (e.g., even terms AP, odd terms GP) | 2, 3, 6, 9, 18, 27, … | Separate rules for sub‑sequences. |
| Digit‑Based | Operate on digits (sum, product, reversal, etc.) | 121, 242, 363, … | Pattern in digits, not values. |
Tip: Never assume the pattern is “only” one of the above; many test series are hybrid.
3. Key Formulae & Their Derivations
| Formula | Symbol | Derivation | Remark |
|---|---|---|---|
| Sum of first (n) terms of AP | (S_n = \dfrac{n}{2}(2a_1 + (n-1)d)) | Pairing first & last: ((a_1+a_n) \times \frac{n}{2}). | (a_n = a_1+(n-1)d). |
| Sum of first (n) terms of GP | (S_n = a_1,\dfrac{1-r^n}{1-r}) (;(r\neq1)) | Geometric series sum: (\sum r^{,k}). | For (r>1), use (r^n-1). |
| Sum of first (n) squares | (\displaystyle \sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}) | Induction or known formula. | Helpful for quadratic series. |
| Sum of first (n) cubes | (\displaystyle \sum_{k=1}^n k^3 = \Big[\frac{n(n+1)}{2}\Big]^2) | Square of triangular numbers. | Useful for series of cubes. |
| Nth term of AP | (a_n = a_1 + (n-1)d) | By definition. | |
| Nth term of GP | (a_n = a_1,r^{,n-1}) | By definition. | |
| Nth term of Fibonacci | (F_n = \dfrac{\phi^n-\psi^n}{\sqrt5}) ((\phi=\frac{1+\sqrt5}{2}), (\psi=\frac{1-\sqrt5}{2})) | Binet’s formula. | Not usually needed in reasoning tests. |
4. General Approach to Solving Series Problems
| Step | Action | What to Look For | Common Pitfall |
|---|---|---|---|
| 1 | Write down the given terms clearly. | Identify if terms are integers, fractions, or symbols. | Ignoring negative signs or decimal places. |