x = 1 + 1 2 1 + 2 ! 1 ( 4 ) 1 2 2 1 + 3 ! 1 ( 4 ) ( 7 ) 1 2 3 1 + 4 ! 1 ( 4 ) ( 7 ) ( 1 0 ) 1 2 4 1 + ⋯
Find the value of 3 x 3 .
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Now why is the question in french?
Sir @Calvin Lin can you please change the wording of the problem to English. It was a beautiful problem and I want everyone else to try it.
I guess, the problem statement would something like - If x = 1 + 1 2 1 + 2 ! 1 ( 4 ) 1 2 2 1 + 3 ! 1 ( 4 ) ( 7 ) 1 2 3 1 + … Then find the value of 3 x 2
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Vous possédez de bonnes compétences en effet d'observation :) Je ne aurais jamais fait le chemin que vous avez fait .
Au fait ,je suppose que la question aurait été mieux si ce était comme suit :
x = 1 + 1 2 1 + 2 ! 1 ( 2 ) 1 2 2 1 + 3 ! 1 ( 2 ) ( 4 ) 1 2 3 1 + 4 ! 1 ( 2 ) ( 4 ) ( 7 ) 1 2 4 1 + …
Que pensez-vous ?
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fait amusant : Jetez un oeil à la date de cette solution posté ...
(Google Translate is still so incapable)
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We might think of applying the Generalized Binomial Theorem , but it can't be applied directly.
So, need to do some modifications with the series so that we can easily see some binomial pattern.
x = 1 + 1 2 1 + 2 ! 1 ( 4 ) 1 2 2 1 + 3 ! 1 ( 4 ) ( 7 ) 1 2 3 1 + 4 ! 1 ( 4 ) ( 7 ) ( 1 0 ) 1 2 4 1 + …
We'll break 1 2 x ( x is any arbitrary power of 12) into 4 x × 3 x and take 3 x in the numerator. Observe carefully -
x = 1 + 3 1 × 4 1 + 2 ! 3 1 × 3 4 4 2 1 + 3 ! 3 1 × 3 4 × 3 7 4 3 1 + …
x = 1 + 3 − 1 × 4 − 1 + 2 ! 3 − 1 × ( 3 − 1 − 1 ) 4 2 1 + 3 ! 3 − 1 × ( 3 − 1 ) × ( 3 − 1 − 2 ) 4 3 − 1 + …
Now, this is nothing but expansion of x = ( 1 − 4 1 ) 3 − 1 ⇒ ( 4 3 ) 3 − 1 ⇒ ( 3 4 ) 3 1
Hence, x = ( 3 4 ) 3 1 . Therefore, 3 x 3 = 4