if-f-f-f-f-x-n-times-2x-1-find-f-1-

Question Number 164299 by mr W last updated on 15/Jan/22

i f \underset{n t i m e s}{f (f (f (\dots f (x) \dots)))} = 2 x + 1,

f i n d f (1) = ?

Answered by ajfour last updated on 16/Jan/22

y_{1} = a x + b

y_{2} = a (a x + b) + b

y_{3} = a^{2} (a x + b) + a b + b

y_{n} = a^{n} x + (a^{n - 1} + a^{n - 2} + \dots + a + 1) b

= 2 x + 1

\Rightarrow a^{n} = 2

(\frac{a^{n} - 1}{a - 1}) b = 1

\Rightarrow b = a - 1 = 2^{1 / n} - 1

y_{1} (1) = f (1) = a + b = 2 a - 1

\Rightarrow f (1) = 2^{\frac{n + 1}{n}} - 1

e . g . s a y n = 5

y_{5} = a^{5} x + b (\frac{a^{5} - 1}{a - 1}) = 2 x + 1

\Rightarrow a^{5} = 2; b = 2^{1 / 5} - 1

y_{1} = (2^{1 / 5}) x + 2^{1 / 5} - 1

y_{1} (1) = 2^{6 / 5} - 1 (s e e b l u e f o r m u l a) .

Commented by mr W last updated on 16/Jan/22

p e r f e c t s i r!

Commented by Tawa11 last updated on 16/Jan/22

Great sir

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