$$ S = 1 + 2 +...+ n = \sum_{k = 1}^{n}k =\frac {n(n+1)}{2}, \space \forall n \in \N $$
Supongamos $q \in \R$
$$ Q = 1 + q^2 + ... + q^n = \sum_{k=0}^n q^k=\begin{cases} \frac{q^{n + 1} -1}{q -1} &\text{cuando } q \not =1 \\ n + 1 &\text{cuando } q = 1 \end{cases}, \space \forall n \in \N $$
$$ \sum_{i=1}^{n+1} a_i = \sum_{i=1}^n a_i + a_{n+1}, \space \forall n \in \N \\ \text{con } \sum_{i=1}^1 a_i = a_1 $$
Propiedades
$$ \bigg( \sum_{k=1}^n a_k\bigg) + \bigg( \sum_{k=1}^n b_k\bigg) = \sum_{k=1}^n (a_k + b_k) $$
$$ \sum_{k=1}^n (c \cdot a_k) = c \cdot \sum_{k=1}^n a_k $$
$$ a_1 \cdot a_2 \cdot... \cdot a_n = \prod_{i=1}^n a_i $$
Propiedades
$$ \prod_{i=1}^n i = 1 \cdot 2 \cdot... \cdot n = n! $$
$$ \prod_{i=1}^n c = \underbrace{c \cdot c \cdot... \cdot c}_{n \text{ veces}} = c^n $$
$$ \bigg(\prod_{k=1}^n a_k\bigg) \cdot \bigg(\prod_{k=1}^n b_k\bigg)= \prod_{k=1}^n (a_k \cdot b_k) $$
$$ \prod_{k=1}^n (c \cdot a_k) = \bigg(\prod_{k=1}^n c\bigg) \cdot \bigg(\prod_{k=1}^n a_k\bigg) = c^n \cdot \prod_{k=1}^n a_k $$
Sea $p(n)$ una proposición sobre $\N$