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Geometry Level 4

∑ r = 1 100 arcsin ⁡ ( 1 r 2 + 1 ⋅ r 2 + 2 r + 2 ) \sum_{r=1}^{100}\arcsin \left({\frac{1}{\sqrt{r^2+1} \cdot \sqrt{r^2+2r+2}}}\right) .

Find the value of the summation above.

Details and Assumptions:

  • Here the answer is in radians .
  • You can use a calculator for the last step.
  • If any moderator can fix the latex, please do so and then delete this line. Thanks!


The answer is 0.7754.

Md Zuhair by Md Zuhair , 20, India

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2 solutions

Suhas Sheikh
May 28, 2018

Convert the given general expression into a telescopic sum of the arcsines Then just plug in values with the calculator To get what is needed

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@Saitama Lawliet Wow.....would you please care to elaborate a bit??

Aaghaz Mahajan - 3 years ago

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What I mean is this Try converting the given expression of arcsin Into arcsinx - arcsiny Where x and y are consecutive terms of the sequence You'll thus end up with a telescopic sequence Allowing you to evaluate the result with ease

Suhas Sheikh - 3 years ago
Kumar Krish
Nov 1, 2019

I'm giving you only a hint See (r^2+r+1)^2 +1 = (r^2+1) (r^2+2r+2) With the help of this convert it into telescopic series of arctan and then you will get answer as Arctan(101) -π/4👍

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