Too complicated
Consider the recurrence relation f n + 1 ( x ) = x + 2 f n ( x ) for n = 1 , 2 , 3 , … and f 1 ( x ) = 3 x .
Find the sum of all possible roots of x = f n ( x ) for all positive integer n .
The answer is 3.
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1 solution
Write easyly 2 1 ( x 2 − x ) = x following x=0 and x=3.
@Prakhar Bindal - Try this one out
@Pi Han Goh - You changed my problem language now what about my solution. It does not go well with the new language
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It's wrong from the start. You have only shown that the answer is 3 when n → ∞ .
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It was correct according to previous language
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@Anubhav Tyagi – No, it's not. Tell me how you've shown (in your solution) that the answer is 3 regardless of the value of n .
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@Pi Han Goh – You may check by putting for n= any integer
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@Anubhav Tyagi – Are you saying you've done proof by exhaustion for all positive integers n ?
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@Pi Han Goh – Yes any doubts
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@Anubhav Tyagi – First of all, you have not shown that in your solution. And second of all, you can't perform "proof by exhaustion" for an infinitely many number of cases because it's an impossible task.
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@Pi Han Goh – It has got 3 upvotes which means that people might have understood it.
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@Anubhav Tyagi – I could upvote it too, does it mean that your solution is correct?
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@Pi Han Goh – You won't cause you will never learn to appreciate
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@Anubhav Tyagi – This is a very fruitful discussion.
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@Pi Han Goh – What do you mean?
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@Anubhav Tyagi – He is being sarcastic.
Please be respectful of moderators, they have to take decisions in behalf of the community, which is not an easy task.
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@Agnishom Chattopadhyay – Are you saying that I am being sarcastic?
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This is a very fruitful discussion
That is a sarcastic comment.
⇒ x = x + 2 x ⇒ x = 0 , 3 x = f n ( x ) = x + 2 f n − 1 ( x ) = x + 2 x + 2 f n − 2 ( x ) Hence we write as: x = x + 2 x + 2 x + 2 ⋯ + 2 3 x ( n times ) x = x + 2 x + 2 x + 2 ⋯ + 2 x + 2 x ( n times ) Replace underlined x from equation (1) x = x + 2 x + 2 x + 2 ⋯ + 2 x + 2 x ( 2 n times ) Replace underlined x from equation (1) x = x + 2 x + 2 x + 2 ⋯ + 2 x + 2 x ( 3 n times ) Replace underlined x from equation (1) x = x + 2 x + 2 x + 2 ⋯ + 2 x + ⋯ ( k → ∞ lim k times ) ( 1 )