Series – Advanced Study Notes (MIT‑Level)
Subject: Reasoning Ability – Sikkim PSC
Audience: Advanced high‑school/undergraduate students
A series in reasoning ability is a finite or infinite sequence of numbers (or symbols) generated by a rule that can be detected through pattern recognition, algebraic manipulation, or logical inference. In competitive exams, the series is usually short (≤ 10 terms) and asks for the next term, missing term, or rule governing the sequence.
1. Foundations
1.1 Definition
A sequence (or series) is an ordered list ( {a_n}_{n=1}^\infty ) where each term ( a_n ) is a well‑defined mathematical object (number, symbol, etc.).
- Finite series: ( a_1, a_2, \dots, a_k ).
- Infinite series: ( a_1, a_2, \dots ).
1.2 Notation
- (a_n) – nth term.
- (d_n = a_{n+1} - a_n) – first difference.
- ( \Delta^k a_n ) – kth finite difference.
1.3 Basic Tools
| Tool | What it does | Typical use |
|---|---|---|
| Difference table | Computes successive differences | Detects polynomial degree |
| Ratio table | Computes successive ratios | Detects geometric progression |
| Recurrence relation | Expresses (a_n) in terms of previous terms | Fibonacci, Lucas, etc. |
| Generating function | Encodes sequence as a formal power series | Solving linear recurrences |
| Polynomial interpolation | Finds polynomial that fits given terms | Predicting next term when degree ≤ k-1 |
2. Classic Series Types
| Type | General form | Key properties | Common exam patterns |
|---|---|---|---|
| Arithmetic (AP) | ( a_n = a_1 + (n-1)d ) | Constant difference (d) | 2,5,8,11,? |
| Geometric (GP) | ( a_n = a_1 r^{,n-1} ) | Constant ratio (r) | 3,6,12,24,? |
| Harmonic | ( a_n = \frac{1}{a_1 + (n-1)d} ) | Reciprocal of AP | 1, 1/2, 1/3,? |
| Quadratic progression | ( a_n = An^2 + Bn + C ) | Second differences constant | 1,4,9,16,? |
| Cubic / Higher‑degree | Polynomial of degree (k) | (k)th differences constant | 1,8,27,64,? |
| Exponential series | ( a_n = c^n ) | Growth/decay | 2^1,2^2,2^3,? |
| Factorial | ( a_n = n! ) | Super‑exponential | 1,2,6,24,? |
| Fibonacci‑type | ( a_n = a_{n-1} + a_{n-2} ) | Linear recurrence | 1,1,2,3,5,? |
| Alternating series | Involves sign changes | 1,-1,1,-1,? |
3. Rigorous Theory Behind Common Series
3.1 Arithmetic Progression (AP)
Theorem 1
For an AP with first term (a_1) and common difference (d), the nth term is
[
a_n = a_1 + (n-1)d.
]
Proof.
By definition ( a_2 = a_1 + d, a_3 = a_2 + d = a_1 + 2d,\dots). Induction gives the formula.
Sum of first (k) terms
[ S_k = \frac{k}{2}(a_1 + a_k) = \frac{k}{2}\left[2a_1 + (k-1)d\right]. ]
3.2 Geometric Progression (GP)
Theorem 2
For a GP with first term (a_1) and ratio (r \neq 1), the nth term is
[
a_n = a_1 r^{,n-1}.
]
Proof – By induction using (a_{n+1} = a_n r).
Sum of first (k) terms
[ S_k = a_1 \frac{1-r^k}{1-r}, \quad r\neq 1. ]