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First off, notice that \( b^{-1} = ac , a^{-1} = bc, c^{-1} = ab \).

Then,

\( \dfrac{1}{a + b^{-1} + 1} = \dfrac{1}{a + ac + 1} \)

\( \dfrac{1}{b + c^{-1} + 1} = \dfrac{1}{b + ab + 1} = \dfrac{abc}{b + ab + abc} = \dfrac{ac}{1+a+ac}\)

\( \dfrac{1}{c + a^{-1} + 1} = \dfrac{1}{c + bc + 1} = \dfrac{abc}{c+cb + abc} = \dfrac{ab}{1+b+ab} = \dfrac{ab}{abc + b + ab} = \dfrac{a}{1+ a + ac} \)

Add them all up and you'll get the answer. Siddhartha Srivastava · 2 years ago

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@Siddhartha Srivastava thank's very much for your answer. I didn't think like that before. See you at other question. Raju Pratama · 2 years ago

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Please mention the source of this problem. @Raju Pratama Swapnil Das · 2 years ago

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@Swapnil Das it is my homework from my math course. Raju Pratama · 2 years ago

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@Raju Pratama O! See my inductive process! Swapnil Das · 2 years ago

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@Swapnil Das but I don't get the complete process from you Raju Pratama · 2 years ago

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@Raju Pratama Just put my values of a, b and c in your expression, this is called induction! Swapnil Das · 2 years ago

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@Swapnil Das how can you get the value of a, b, and c? Raju Pratama · 2 years ago

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@Raju Pratama By random thinking bro! You may take other's too! like 4, 1/4 and 1! Swapnil Das · 2 years ago

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@Swapnil Das what induction is? Raju Pratama · 2 years ago

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@Raju Pratama See Wikipedia on Mathematical Induction. The explanation given by Swapnil is incorrect. Karthik Venkata · 2 years ago

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@Raju Pratama Well, it's kinda technique used to prove mathematical theorems by taking random values, rather than actually proving them by expressions and hard simplifications! Swapnil Das · 2 years ago

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@Swapnil Das This is the worst misconception of Mathematical Induction :/ ! Karthik Venkata · 2 years ago

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@Karthik Venkata Hey, I saw my mistypes! Actually I meant Inductive reasoning! Sorry for that! Swapnil Das · 2 years ago

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@Swapnil Das Ya, No problem. but by the way what the different between Inductive reasoning and Mathematical Induction? Raju Pratama · 2 years ago

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@Raju Pratama Well,... I donno how should I explain, BTW angry on my math teacher, It was he who misconceptualized my brain! Swapnil Das · 2 years ago

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@Swapnil Das Oooh, I'm sorry to hear that. Raju Pratama · 2 years ago

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@Karthik Venkata O, I really don't know about it before. Thank's a lot ffor your explaination Karthik Venkata Raju Pratama · 2 years ago

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@Swapnil Das but, do you have a manual way? because my teacher asked me to write the process. Raju Pratama · 2 years ago

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@Raju Pratama Ya, but difficult to edit! Swapnil Das · 2 years ago

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@Swapnil Das if I express 4, 1/4, and 1. I don't get the answer. how about other values? Raju Pratama · 2 years ago

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@Raju Pratama U don't ! Lemme try! Swapnil Das · 2 years ago

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@Swapnil Das how if a = -1, b = -1, and c = 1 ? Raju Pratama · 2 years ago

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@Raju Pratama YA! Swapnil Das · 2 years ago

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If U give me sufficient time, I can provide you the complete process by trying myself. Swapnil Das · 2 years ago

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Well by what time do you want the answer? Swapnil Das · 2 years ago

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@Swapnil Das at 20.45 Raju Pratama · 2 years ago

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@Raju Pratama What's time now? ( Your profile suggests that U live in Indonesia) Swapnil Das · 2 years ago

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@Swapnil Das well, maybe you can solve it in 15 minutes Raju Pratama · 2 years ago

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@Swapnil Das 20.30 Raju Pratama · 2 years ago

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@Raju Pratama My goodness! Swapnil Das · 2 years ago

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Please replace my induction with inductive reasoning, I was mistakened as my teacher gave me a wrong explanation. Thanks! Swapnil Das · 2 years ago

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Did U get it? Swapnil Das · 2 years ago

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@Swapnil Das no, I don't, but would you like to explain the complete process to me? Raju Pratama · 2 years ago

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@Raju Pratama Think of some numbers a,b,c such that abc=1, I thought and found out! Then put them in the expression! Swapnil Das · 2 years ago

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@Raju Pratama It's one! I did it by induction! take a =2, b=1/2 and c = 1 ! Swapnil Das · 2 years ago

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@Swapnil Das Lol how can it be called induction Swapnil Das ? I think you are confused between Inductive reasoning (guessing) and Mathematical Induction ! See this. Karthik Venkata · 2 years ago

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please comment. Raju Pratama · 2 years ago

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@Raju Pratama I think the answer is one. Swapnil Das · 2 years ago

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@Swapnil Das how the way? Raju Pratama · 2 years ago

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@Raju Pratama I have not tried, just U try to take the following method I always use, abc=1, then a=??, and go on! Swapnil Das · 2 years ago

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@Raju Pratama What's your age bro? Swapnil Das · 2 years ago

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@Swapnil Das 17 years old, how about you? Raju Pratama · 2 years ago

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@Raju Pratama O! I am 14 sir!! Swapnil Das · 2 years ago

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