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TopNewestFirst 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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– Raju Pratama · 2 years ago

thank's very much for your answer. I didn't think like that before. See you at other question.Log in to reply

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

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– Raju Pratama · 2 years ago

it is my homework from my math course.Log in to reply

– Swapnil Das · 2 years ago

O! See my inductive process!Log in to reply

– Raju Pratama · 2 years ago

but I don't get the complete process from youLog in to reply

– Swapnil Das · 2 years ago

Just put my values of a, b and c in your expression, this is called induction!Log in to reply

– Raju Pratama · 2 years ago

how can you get the value of a, b, and c?Log in to reply

– Swapnil Das · 2 years ago

By random thinking bro! You may take other's too! like 4, 1/4 and 1!Log in to reply

– Raju Pratama · 2 years ago

what induction is?Log in to reply

– Karthik Venkata · 2 years ago

See Wikipedia on Mathematical Induction. The explanation given by Swapnil is incorrect.Log in to reply

– Swapnil Das · 2 years ago

Well, it's kinda technique used to prove mathematical theorems by taking random values, rather than actually proving them by expressions and hard simplifications!Log in to reply

– Karthik Venkata · 2 years ago

This is the worst misconception of Mathematical Induction :/ !Log in to reply

– Swapnil Das · 2 years ago

Hey, I saw my mistypes! Actually I meant Inductive reasoning! Sorry for that!Log in to reply

– Raju Pratama · 2 years ago

Ya, No problem. but by the way what the different between Inductive reasoning and Mathematical Induction?Log in to reply

– Swapnil Das · 2 years ago

Well,... I donno how should I explain, BTW angry on my math teacher, It was he who misconceptualized my brain!Log in to reply

– Raju Pratama · 2 years ago

Oooh, I'm sorry to hear that.Log in to reply

– Raju Pratama · 2 years ago

O, I really don't know about it before. Thank's a lot ffor your explaination Karthik VenkataLog in to reply

– Raju Pratama · 2 years ago

but, do you have a manual way? because my teacher asked me to write the process.Log in to reply

– Swapnil Das · 2 years ago

Ya, but difficult to edit!Log in to reply

– Raju Pratama · 2 years ago

if I express 4, 1/4, and 1. I don't get the answer. how about other values?Log in to reply

– Swapnil Das · 2 years ago

U don't ! Lemme try!Log in to reply

– Raju Pratama · 2 years ago

how if a = -1, b = -1, and c = 1 ?Log in to reply

– Swapnil Das · 2 years ago

YA!Log in to reply

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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– Raju Pratama · 2 years ago

at 20.45Log in to reply

– Swapnil Das · 2 years ago

What's time now? ( Your profile suggests that U live in Indonesia)Log in to reply

– Raju Pratama · 2 years ago

well, maybe you can solve it in 15 minutesLog in to reply

– Raju Pratama · 2 years ago

20.30Log in to reply

– Swapnil Das · 2 years ago

My goodness!Log in to reply

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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– Raju Pratama · 2 years ago

no, I don't, but would you like to explain the complete process to me?Log in to reply

– Swapnil Das · 2 years ago

Think of some numbers a,b,c such that abc=1, I thought and found out! Then put them in the expression!Log in to reply

@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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this. – Karthik Venkata · 2 years ago

Lol how can it be called induction Swapnil Das ? I think you are confused between Inductive reasoning (guessing) and Mathematical Induction ! SeeLog in to reply

please comment. – Raju Pratama · 2 years ago

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– Swapnil Das · 2 years ago

I think the answer is one.Log in to reply

– Raju Pratama · 2 years ago

how the way?Log in to reply

– Swapnil Das · 2 years ago

I have not tried, just U try to take the following method I always use, abc=1, then a=??, and go on!Log in to reply

– Swapnil Das · 2 years ago

What's your age bro?Log in to reply

– Raju Pratama · 2 years ago

17 years old, how about you?Log in to reply

– Swapnil Das · 2 years ago

O! I am 14 sir!!Log in to reply