Math Problems, Number Theory Problems

Problem of the Day: 3/14/13

What is the sum of the terms in the sequence 35+34+36+37+33+32+38+39+ … +2+1+69+70?

Solution to yesterday’s problem:

We can assume that triangle PQR has integer sides because more information was not specified. This would mean that PQR is a Pythagorean triple with a short side of 9: a 9-12-15. So the hypotenuse of PQR is 15 cm.

Now we’re going to do some more estimating! You can sort of see that the small triangle QUT takes about a third of the hypotenuse. This goes along with our theory about integer lengths because it would make the hypotenuse 5, which leads us to believe that QUT is a 3-4-5 triangle. Also, the larger triangle RST takes up two-thirds of the hypotenuse, or 10 cm. This suggests that RST is a 6-8-10 triangle. So let’s show the triangle again with the lengths labeled:

20130314-091142.jpg

The length we actually want to find is US, conveniently the only segment we don’t know. We can use the Pythagorean theorem for triangle STU.

4^2+6^2=x^2

16+36=x^2

52=x^2

\sqrt{52}=x

Simplifying the radical,

\sqrt{4\times13}=x

\sqrt{4}\times\sqrt{13}=x

2\sqrt{13}=x

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