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aritra nandy Grade: 10
        if y=x+1/x+1/x+1/x+............

prove that
note that the + sign in the sum goes only with the denominator and not with the only sum so,
we can
sum it as y=x+1/y
then what should be done??
7 years ago

Answers : (1)

Priyansh Bajaj AskiitiansExpert-IITD
30 Points

Dear Aritra,

Solution:- As you said, we can write the given eq. y = x + 1/(x+ 1/(x+ 1/(x+...))) as y = x+ 1/y. Then, you should differentiate the implicit eq. and evaluate dy/dx. Then, do some manipulation to achieve desired results. You can refer the following solution-

              y = x +1/y.................(1)

              dy/dx = 1 - (1/y2) (dy/dx)

           [1 + (1/y2)] (dy/dx) = 1..................(2)

        From eq. (1), y-1/y = x. On squaring both sides, we get-

 (y-1/y )2= x2 or  y2 + 1/y2 -2= x2 or 1/y2 = (x2 - y2+2)

Putting the value of 1/yin eq. (2), we get-

  [1 + (x2 - y2+2)] (dy/dx) = 1

(x2 - y2+3) (dy/dx) = 1 

Hence proved.                        

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All the best Aritra!!!

Askiitians Experts
Priyansh Bajaj

7 years ago
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