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--- tanszek:oktatas:techcomm:information_-_basics:sciences [2025/09/15 17:32] – [What is science?] knehez
+++ tanszek:oktatas:techcomm:information_-_basics:sciences [2025/09/15 17:52] (current) – [Inductive Sciences] knehez
@@ Line 9: / Line 9: @@
 Science, among our historically established forms of social consciousness, is distinguished and emphasized by the following characteristics.
-  * It possesses high-reaching concepts or logical tools to formulate or express broad, general or universal **principles** or **laws** (e.g. gravity, axioms, [[https://en.wikipedia.org/wiki/Maxwell%27s_equations|Maxwell's equations]])
+  * It possesses high-reaching concepts or logical tools to formulate or express broad, general or universal **principles** or **laws** or **axioms** (e.g. gravity, axioms, [[https://en.wikipedia.org/wiki/Maxwell%27s_equations|Maxwell's equations]])
   * It can describe the objective **conditions** under which these principles or laws will prevail.
@@ Line 80: / Line 80: @@
 $$ 1 + 2 + \cdots + k = \frac{k(k+1)}{2}. $$
+Then for \(n = k+1\):
+$$ 1 + 2 + \cdots + k + (k+1) = \frac{k(k+1)}{2} + (k+1). $$
 **Simplify**:
-$$ \frac{k(k+1) + 2(k+1)}{2} = \frac{(k+1)(k+2)}{2}. $$
-Thus, the formula also holds for \( n = k+1 \).
+$$ \frac{k(k+1)}{2} + (k+1) = \frac{k(k+1) + 2(k+1)}{2} = \frac{(k+1)(k+2)}{2}. $$
+Thus, the formula also holds for \( n = k+1 \). Thus, by mathematical induction, the formula holds for all \(n\in\mathbb{N}\)
 https://en.wikipedia.org/wiki/Mathematical_induction