- #1
Proving an equality using induction proof not working
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SUMMARY
This discussion focuses on proving an equality using two-variable induction proofs involving the expression ##\prod\limits_{n=1}^{k}n(2k+2-2n)=2^k(k!)^2##. The inductive hypothesis is established for a specific case, and participants suggest that the induction step should be approached by assuming the hypothesis holds for ##k=m## and then proving it for ##k=m+1##. Additionally, an alternative approach is proposed, recommending direct simplification of the product instead of relying solely on induction.
PREREQUISITES- Understanding of mathematical induction, particularly for two variables.
- Familiarity with factorial notation and properties.
- Knowledge of product notation and simplification techniques.
- Basic algebraic manipulation skills, including the use of parentheses in expressions.
- Study the principles of mathematical induction, focusing on two-variable cases.
- Learn about simplifying products in algebra, particularly in combinatorial contexts.
- Explore alternative proof techniques that can complement induction, such as direct computation.
- Review common pitfalls in algebraic expressions, particularly regarding the use of parentheses.
Students studying mathematics, particularly those tackling proofs in combinatorics or algebra, as well as educators looking for effective teaching strategies in induction proofs.
- #2
pizzasky
- 172
- 2
Our inductive hypothesis is ##\prod\limits_{n=1}^{k}n(2k+2-2n)=2^k(k!)^2##, for some ##k\in\{1,2,...\}##.
We want to show that ##\prod\limits_{n=1}^{k+1}n(2(k+1)+2-2n)=2^{k+1}((k+1)!)^2##.
We want to show that ##\prod\limits_{n=1}^{k+1}n(2(k+1)+2-2n)=2^{k+1}((k+1)!)^2##.
- #3
SammyS
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tony700 said:
That's what you're to prove.
I think it's clearer if you do the induction step as follows.
Assume that ##\displaystyle \ \prod_{n=1}^{k}n(2k+2-2n)=2^k(k!)^2 \ ## is true for ##\ k=m\ ## for some ##m>0##. Then show that it's true for ##k = m+1##. You must replace every ##k## with ##m## or ##m+1## as appropriate.
Note: In the jpeg image that you showed, you needed to have extra parentheses in a number of places.
- #4
Ray Vickson
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Are you absolutely required to use induction? If not, just writing out the product directly and simplifying is by far the easiest way to do the problem.tony700 said:
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