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Nice one andrew. Enjoyed it. Thanks for putting it up.

None of the answers are complete.

a = 0 also is a solution. a = 0 and a=4/9 are all the solutions. This is just the kind of mistake one should avoid making in an exam.

Getting lots of errors in using tex. .

Both the first proof (with infinities) and the “intuitive one” are the same. Any explanations to this from the community?

Here’s a hint: both the proofs are related to the formula for the sum of the geometric series.

The mind is not a vessel to be filled, it is a fire to be kindled. ~ Plutarch

Easy reading is damned hard writing. ~ Nathaniel Hawthorne

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. ~ G. H. Hardy

I have a theory on why kids (and adults when they grow up) don’t like math. Here’s the theory.

Assumption: When you get the right answer in a math problem, you feel great. Conversely, when you get the wrong answer, you feel horrible.

So the kids who get the right answer, feel math is wonderful, and this feeling builds on itself. Over time learners are able to appreciate the beauty of math, and they begin to love math.

But there are so many problems, that everyone does get the wrong answer sometimes, and that is where negative emotions for math begin building up.

Like lechuck says, eventually students will feel stupid. My group theory prof (a japanese prof called Harada) used to say…math is very strange: if you know something, its easy, if you don’t, it very hard.

Teachers too don’t realise this. Notice how math teachers have a tendency to put down students getting the wrong answer. Notice how they reward students when they get the problem quickly.

Actually, how quickly you get a math problem is not so important…how much time you spend trying is more important. If you have to think about a puzzle for a long time and then getting the solution gives a great feeling. Its the same as making a big effort on a painting, or practicing for long hours to give a good concert, or finding the bug in your code after worrying over it all night. Those are the feelings that a teacher has to generate and communicate.

And teachers have to teach students how to overcome the pain of not being able to do the problem, or getting the right answer.

Anyhow, by and large, if we teachers give opportunities to students to succeed in getting the right answer, and teach them to deal with getting it wrong sometimes, I expect students will begin liking math.

What say?

PS. Sorry for the delayed responses…somehow I just saw a mail today that there has been an update on this discussion. Maybe the discussions can also be reported in the inbox?

So we can write in tex now? Thats great.