Ideas from Classroom Teachers for
Mathematical Induction, Sequences, and Series
If sigma-notation is new to students at this stage, a day’‘s worth of just that topic would be a good idea.
Depending on the students you have, they may either take up the challenge of deriving formulas for sums or they may completely downshift. This is a unit where it would be useful to have a number of backup plans. This is also material that teachers are likely to have not seen in a long time (possibly not since college).
Induction could be motivated by exploring the sums of the first n odd numbers, discovering a pattern (the sum of the first n odd numbers is n2), and recognizing a need to prove this conjecture.
An effective counter-example for not generalizing based on a few examples: Is n2 + n + 41 prime for all natural numbers n? Yes, for n up to 40, but not for n = 41.
Induction could be used to prove the general arithmetic and geometric series formulas.
Applications of Pascal’‘s Triangle, the Binomial Theorem, and Fibonacci numbers are plentiful and easy to find.
Pascal’‘s Triangle could be explored as a way to find the binomial expansion of (a + b)n for small values of n. Connections between binomial coefficients, Pascal’‘s Triangle, and combinations (nCr) could be considered. Students could then prove the Binomial Theorem by induction.
Example of divisibility: Prove that 7n - 1 is divisible by 6 for all natural numbers n.
Example of inequality: Prove n < 2n for all natural numbers n.
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