Quantitative Aptitude – Tables
Program: Tripura Patwari / Lekhpal
Level: MIT‑style (Advanced university)
1. What Is a “Table”?
| Concept | Definition |
|---|---|
| Numerical table | A two–dimensional arrangement of numbers that lists values of one or more functions for a set of arguments. |
| Use in aptitude | Quickly look up or interpolate values of powers, factorials, logarithms, combinatorial numbers, etc., without full calculation. |
| Types | 1. Multiplication / Power tables 2. Factorial / Binomial tables 3. Logarithmic tables 4. Trigonometric tables 5. Combinatorial tables (e.g., Stirling, Bell numbers). |
Why are tables “advanced”?
They encode finite‑difference and polynomial structures that allow us to reconstruct entire functions from a few sampled values. They also serve as early examples of data compression and pre‑computation in algorithmic design.
2. Constructing Tables from First Principles
2.1. Defining the Function
Let ( f : \mathbb{Z} \to \mathbb{R} ) be a function we wish to tabulate (e.g., (f(n)=2^n), (f(n)=\log_{10}n)).
We choose a domain (D={n_0, n_0+1, \dots, n_0+N-1}).
2.2. Finite Differences
For equally spaced integer arguments, the forward difference operator (\Delta) is defined by
[ \Delta f(n) = f(n+1)-f(n). ]
Higher‑order differences obey
[ \Delta^k f(n) = \Delta^{k-1} f(n+1) - \Delta^{k-1} f(n). ]
Key property: If (f) is a polynomial of degree (d), then (\Delta^{d+1} f(n)=0) for all (n).
Example:
(f(n)=n^2)
(\Delta f(n)=2n+1)
(\Delta^2 f(n)=2)
(\Delta^3 f(n)=0).
2.3. Newton Forward / Backward Interpolation
Given (f(n_0), f(n_0+1),\dots,f(n_0+N-1)), we can reconstruct (f) for any integer (m) in the range using
[ f(m) = f(n_0) + \frac{m-n_0}{1!}\Delta f(n_0)+\frac{(m-n_0)(m-n_0-1)}{2!}\Delta^2 f(n_0)+\cdots ]
This is equivalent to the Lagrange interpolation polynomial but is more efficient for integer arguments.
2.4. Generating Recursive Tables
Often the values satisfy a recurrence relation:
[ f(n+1)=g(n,f(n),f(n-1),\dots). ]
For example, factorial: (n! = n \times (n-1)!).
Binomial coefficients satisfy Pascal’s rule: (\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}).
Recursive generation is memory‑efficient and numerically stable because each entry is computed from a few previous entries.
3. Advanced Theory Behind Tables
3.1. Polynomial Approximation & Interpolation
-
Lagrange basis polynomials
[ L_k(x)=\prod_{\substack{j=0\ j\neq k}}^{N-1}\frac{x-j}{k-j} ] The interpolating polynomial is (P(x)=\sum_{k=0}^{N-1}f(k)L_k(x)). -
Error term (for (f) sufficiently smooth)
[ f(x)-P(x)=\frac{f^{(N)}(\xi)}{N!}\prod_{k=0}^{N-1}(x-k), \quad \xi\in[n_0,n_0+N-1]. ] In tables we usually evaluate at integer points, thus the error term vanishes if (f) is a polynomial of degree (\le N-1).
3.2. Generating Functions
Many combinatorial tables are succinctly encoded by generating functions:
- Binomial coefficients: ((1+x)^n = \sum_{k=0}^n \binom{n}{k}x^k).
- Catalan numbers: (C(x)=\