Mathematical Foundations in Coding
Mathematical Foundations in Coding
Gaussian Formula
$$ 1 + 2 + 3 + … + n = \frac{n(n+1)}{2} $$
$$ (1 + 2 + 3 + … + n) \newline + (n + n-1 + n-2 + … + 1) \newline = (n+1) + (n+1) + … + (n+1) \newline = n(n+1) $$
Pascal’s Rule
$$ C_{n+1}^k = C_n^k + C_n^{k-1} \newline C_n^k = \frac{n!}{k!(n-k)!} \newline =\frac{(n-1)!}{k!(n-1-k)!} × \frac{n}{n-k} \newline =C_{n-1}^k × \frac{n}{n-k} $$
$$ \sum_{j=0}^{k-1} C_{i+1}^{j} \newline =\sum_{j=0}^{k-1} (C_{i}^{j} + C_{i}^{j-1}) \newline =\sum_{j=0}^{k-1} C_{i}^{j} + \sum_{j=0}^{k-1} C_{i}^{j-1} \newline =\sum_{j=0}^{k-1} C_{i}^{j} + \sum_{j=0}^{k-2} C_{i}^{j} \newline =\sum_{j=0}^{k-1} C_{i}^{j} + \sum_{j=0}^{k-1} C_{i}^{j} - C_{i}^{k-1} \newline =2\sum_{j=0}^{k-1} C_{i}^{j} - C_{i}^{k-1} \newline $$
Last modified on 2025-01-24