maths.free › Discrete Math & Logic › Sums and induction
Sums and induction
Closed forms for sums, and how induction proves them.
Σ k = n(n+1)/2 is the formula Gauss found at school; Σ k² and Σ k³ have closed forms too. Induction proves them: check n = 1, then show that if the formula holds for n it holds for n + 1.
Exemple résolu · sum of k for k = 1 to n
Étape par étape
- \sum_{k=1}^{n} k
Write the sum out.
- 1 + 2 + 3 + 4 + \cdots
The first few terms.
- = \frac{n \left(n + 1\right)}{2}
Closed form (a standard summation formula, provable by induction).
Révèle la réponse
À vous Choisissez un problème, ou tapez ou dessinez le vôtre. Chaque étape, une image, la réponse cachée jusqu'à ce que vous le demandiez.
Un compte gratuit ajoute des notes sur chaque leçon, un enregistrement de ce que vous avez terminé, vos problèmes résolus en un seul endroit, et un tuteur que vous pouvez demander sur cette page. Les maths eux-mêmes sont ouverts à tout le monde, signé ou non.
Inscription ConnexionSymboles utilisés ici
Appuyez sur n'importe quel symbole pour la définition complète, une image et ce que signifie chaque lettre.
Comment faire: Sums and induction
- Write the sum out.
- The first few terms.
- Closed form (a standard summation formula, provable by induction).
Questions posées par les gens
What makes mathematics "discrete"?
It deals with separate, countable objects (integers, graphs, statements) rather than continuous quantities. No limits, no infinitesimals; instead induction, counting and logic.
How does a proof by induction work?
Show the statement for the first case, then show that whenever it holds for n it holds for n + 1. Like dominoes: the first falls, and each knocks over the next.
What order should I take these lessons in?
Logic and proof first, because every later lesson proves things. Then induction, sets, relations and functions, counting, recurrences, graphs, Boolean algebra and automata, and finally algorithms and complexity, which use nearly everything before them.
Is discrete mathematics the same as the maths of computer science?
Largely. Data structures are graphs and trees, correctness arguments are induction, running times are recurrences and sums, circuits are Boolean algebra, and what a computer can do efficiently is the subject of the last lessons. It is also ordinary mathematics in its own right.
Do I need calculus for this course?
No. School algebra is enough. A little familiarity with limits helps when comparing growth rates, and the lesson on asymptotic analysis explains the one limit fact it uses.
Plus en Discrete Math & Logic
Truth tablesProof by inductionAlgorithms and growth of functionsPredicate logic and quantifiersDirect proof, contrapositive and contradictionStrong induction and the well-ordering principleRelations: equivalence relations and partial ordersFunctions and cardinalityCounting: rules, choices and bijectionsThe pigeonhole principleSolving linear recurrence relationsGraphs: degrees, paths and connectivityTrees and spanning treesEuler and Hamilton paths