Math, asked by Anonymous, 1 year ago

i needed answer of this question ASAP!!

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Sayar: hmm..I will try..

JinKazama1: I think by inverse it goes to 2 . By tan inverse property

JinKazama1: It seems like manipulation of denominator of tan angles

JinKazama1: Is answer 1?

Sayar: yes..

JinKazama1: How can you sure?

Sayar: 0.5+ 1÷3 + 1÷7 + 1÷43 +.... = 1

JinKazama1: OK but I am waiting for one who asked the question

JinKazama1: If@ Robin34 ,confirms my answer then I will show solution as there may be calculation mistake

Sayar: ok then ping me the solution I will also tally with my answer..

Answers

Answered by JinKazama1

Final Answer : 1 .

Steps :
1) From given relation :

$\frac{1}{a(n)} = \frac{a(n) - 1}{a(n + 1) - 1}$

2) Now, write
On solving and substitutions we get,

$\frac{1}{a(n) } = \frac{1}{a(n) - 1} - \frac{1}{a(n + 1) - 1}$

3) Now, do summation
On summation,
This step will be useful:
Lim n to inf 1/ a(n) = 0

For Calculation see Pic

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JinKazama1: There are other 2-3 methods too, if you want inbox me.

Answered by NOBITA01

Another Method :(By Induction)

Here, a(n+1) represents (n+1)th term.
We will first prove the following by induction :
$\frac{1}{a(n + 1) - 1} + summation \: \\ k = 1 \: to \: \: k \: = n \: \: \: \: \: \: \frac{1}{a(k)} \: \: = 1$

Then,
We will substitute value of this summation in which is asked with limit n tends to infinity. .

After Calculation,
Required result = 1.

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