f-Z-Z-f-x-2-f-x-1-f-5-4-find-f-30-

Question Number 148599 by mathdanisur last updated on 29/Jul/21

f : Z \to Z

f (x) = 2 \cdot f (x - 1)

f (5) = 4

f i n d f (30) = ?

Answered by Olaf_Thorendsen last updated on 29/Jul/21

f (x) = 2 f (x - 1) = 2^{2} f (x - 2) = \dots = 2^{25} f (x - 25)

f (30) = 2^{25} f (30 - 25) = 2^{25} f (5) = 2^{25} . 4 = 2^{27}

Commented by mathdanisur last updated on 29/Jul/21

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Answered by Ar Brandon last updated on 29/Jul/21

f (5) = 4, f (4) = 2, f (3) = 1, f (2) = \frac{1}{2}, f (1) = \frac{1}{4}

\Rightarrow f (x) = 2^{x - 3} \Rightarrow f (30) = 2^{27}

Commented by mathdanisur last updated on 29/Jul/21

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Answered by mathmax by abdo last updated on 29/Jul/21

f (n) = 2 f (n - 1) \Rightarrow \frac{f (n)}{f (n - 1)} = 2 \Rightarrow \prod_{k = 6}^{n} \frac{f (k)}{f (k - 1)} = 2^{n - 5} \Rightarrow

\frac{f (6)}{f (5)} . \frac{f (7)}{f (6)} \dots . . \frac{f (n)}{f (n - 1)} = 2^{n - 5} \Rightarrow f (n) = f (5) . 2^{n - 5} \Rightarrow

f (30) = 4 \times 2^{25} = 2^{27}

Commented by mathdanisur last updated on 29/Jul/21

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