Available online
Description
This programme uses cannon balls n layers high in a triangular pyramid to show how proof by mathematical induction can be used to prove results involving the natural numbers.
This programme uses cannon balls n layers high in a triangular pyramid to show how proof by mathematical induction can be used to prove results involving the natural numbers.
| Module code and title: | M101, Mathematics: a foundation course |
|---|---|
| Item code: | M101; 28; 1986 |
| First transmission date: | 07-09-1986 |
| Published: | 1986 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:24:29 |
| + Show more... | |
| Producer: | Pip Surgey |
| Contributors: | Bob Coates; Alan Solomon |
| Publisher: | BBC Open University |
| Keyword(s): | Circle; Conjecture; Counter-example; Divisibility; Induction; Natural numbers; Polygon; Proof; Sum |
| Footage description: | How many areas do you get if you join up n points, taken at random and the edge of a circle? How many cannonballs are there in each layer of a triangular pyramid, of in the top n layers? An equation (...) seems to generate prime numbers, but does it? In this programme, associated with Block VI, Unit 3, students are invited to think about answering these questions finally. All the problems generate conjectures about natural numbers, which leads to a formal method of proof involving natural numbers, namely proof by Mathematical Induction. It turns out that two of the conjectures posed, namely those arising from the circle and prime number problems, are actually false. This is proved by finding counterexamples in each case. The remaining conjectures are proved using Mathematical Induction. It is conjectured that the number of cannon balls in each layer is a triangular number, the formula for which is then proved. Adding the first n triangular numbers gives the total in the top n layers. Mathematical Induction is used to prove that this sum is given by a formula. The chain reaction which is the core of all proofs by Mathematical Induction is described with the aid of a film animation. The programme next considers proving conjectures which do not involve sums. Finally, the principle is extended to consider a more general problem, by conjecturing that the angle sum of an n sided polygon is given by (n-2) pi. The programme makes us of computer animations to establish the conjectures of both the circle and the polygon problem. It should be of interest to students and teachers of mathematics at around Advanced Level. |
| Master spool number: | HOU5370 |
| Production number: | HOU5370 |
| Videofinder number: | 2490 |
| Available to public: | no |
