A blog for enthusiastic math-lovers!

Topic 1: Summations

When I was younger, I always believed that working with problems that looked short and that did not have many scary symbols was easier than dealing with long, complex looking questions. Concepts which especially intimidated me were summations and floor functions. Just by the look of them, they seem s like a mess. However, in reality there are problems in those topics that are very simple!

In summations, often times a concept called telescoping comes into play. Telescoping is when you are able to cancel out most terms in the summation. In summations it is also helpful to sometimes reorganize your problem.

Summations problems come in all different shapes and sizes. The following problem is #14 on the AIME which I spoke about in my previous post.

For each positive integer n, let f(n) =\sum_{k = 1}^{100}\lfloor\log_{10}(kn)\rfloor . Find the largest value of n for which f(n)\le 300 .

Note: \lfloor x\rfloor is the greatest integer less than or equal to.

Please comment if you have any ideas for creative solutions! I will give hints for this problem in later posts! Have fun with this problem!

Also good luck to all of the girls participating in the Math Prize for Girls next weekend!

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Comments on: "Dealing with Intimidating Looking Problems" (1)

  1. The solution to this problem is as follows: we wish to make the summation equal to 300 or less. Since there are 100 terms in the summation, each term should be around 2, 3, or 4. Thus, we realize that n is approximately 100. Using some trial and error, you get when n=109, f(n)=300. Therefore, the answer is 109.

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