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In my task, I've some questions to solve but I am stuck with the following one.

Question: Find the following sum: $\frac{1}{22–1} + \frac{1}{42 –1} + \frac{1}{62 –1} + \dots + \frac{1}{202–1}.$

I tried to find many patterns but couldn't get one to solve it. Like I know the difference is $20$ in each, but there is not any way to find square and make combination, so I am totally stuck with it.

Can anyone guide me with this??

Air Mike
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Naila Akbar
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    Where does this question come from, please? What's the context? – Gerry Myerson Sep 07 '20 at 13:11
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    Wolfram finds $$\sum_{n=1}^{10} \frac{1}{20n+1}=\frac{32423431590702190}{227186523709446609}$$ – K.defaoite Sep 07 '20 at 13:11
  • @K.defaoite provided answer does not match with yours.. – Naila Akbar Sep 07 '20 at 16:28
  • @Null Pointer Could you show us the provided solution then please? – K.defaoite Sep 07 '20 at 18:02
  • Sorry, I don't know...I just googled it and every other link says it's answer is 10/21. (In MCQs portion) no one provided details that's why I am more confused. – Naila Akbar Sep 07 '20 at 19:47
  • By elementary considerations, the denominator must be divisible by $41$, so the sum can't possibly be $10/21$. Either you have stated the problem incorrectly, or everyone who says it's $10/21$ is wrong. So let me repeat my question: where does it come from? What is the context? And let me add, what are these links where it says $10/21$? – Gerry Myerson Sep 08 '20 at 11:08
  • Is there any chance that the sum is really $${1\over2^2-1}+{1\over4^2-1}+\cdots+{1\over20^2-1}$$ – Gerry Myerson Sep 08 '20 at 11:10
  • Nope... @GerryMyerson – Naila Akbar Sep 08 '20 at 14:36
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    I don't know what you want, Null. You say the answer to your question is $10/21$, but several of us have told you that it can't possibly be $10/21$. You refuse to say anything about where the problem came from, or where these alleged links are that say it's $10/21$. What are you expecting to get out of this? The one thing you won't see is a proof that the numbers, the way you have written them, add up to $10/21$, because they don't. So, what do you want? – Gerry Myerson Sep 09 '20 at 01:46
  • Is this where you saw it? https://www.meritnation.com/ask-answer/question/solve-this-q-33-what-is-the-value-of-the-following-exp/real-numbers/12558265 or this? https://www.quora.com/What-is-the-value-of-following-expression-dfrac1-2-2-1-+-dfrac1-4-2-1-+-dfrac1-6-2-1-+-cdots-dfrac1-20-2-1 or this? https://www.toppr.com/ask/question/what-is-the-value-of-the-following-expressionleft-dfrac122-1-right-left – Gerry Myerson Sep 09 '20 at 01:54
  • @GerryMyerson No, not these...I was wrong, 10/21 is not the answer, sorry for confusions and thank you for help. – Naila Akbar Sep 10 '20 at 09:39
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    OK. Now: please, please, please where did the problem come from? – Gerry Myerson Sep 10 '20 at 11:23
  • Actually I got an assignment from university, where this was one the many questions. – Naila Akbar Sep 10 '20 at 12:25
  • So, you've been cheating on your homework, Null? – Gerry Myerson Sep 12 '20 at 00:04
  • Cheating or taking guidance (whatever name it) ..because I was totally stuck on it.. – Naila Akbar Sep 14 '20 at 05:36

2 Answers2

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To get an approximation, write $$S_n=\sum_{k=0}^n \frac{1}{20 k+21}=\frac 1 {20}\sum_{k=0}^n \frac{1}{ k+\frac{21}{20}}\sim \frac 1 {20}\sum_{k=0}^n \frac{1}{ k+1}=\frac 1 {20}H_{n+1}$$ For $n=9$, this would give $$S_{9}=\frac 1 {20}\times\frac{7381}{2520}=\frac{7381}{50400}\approx 0.146448$$ instead of $$S_{9}=\frac{32423431590702190}{227186523709446609}\approx 0.142717$$

Tortar
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Claude Leibovici
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Note that $${2\over n^2-1}={1\over n-1}-{1\over n+1}$$ So $${2\over2^2-1}+{2\over4^2-1}+\cdots+{2\over20^2-1}=1-{1\over3}+{1\over3}-{1\over5}+\cdots+{1\over19}-{1\over21}=1-{1\over21}={20\over21}$$ and we're done (if I've correctly guessed what question OP meant to ask).

Gerry Myerson
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