if-we-let-f-n-n-2-n-1-then-solve-the-equation-for-real-numbers-f-f-f-f-n-1-32-

Question Number 154675 by mathdanisur last updated on 20/Sep/21

if we let

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

then solve the equation for real numbers

f (f (f (f (n)))) = \frac{1}{32}

Answered by MJS_new last updated on 20/Sep/21

n^{2} + n - 2 k = 0 \land n > 0 \Rightarrow n = \frac{\sqrt{8 k + 1} - 1}{2}

g (k) = \frac{\sqrt{8 k + 1} - 1}{2}

g (g (k)) = \frac{\sqrt{4 \sqrt{8 k + 1} - 3} - 1}{2}

g (g (g (k))) = \frac{\sqrt{4 \sqrt{4 \sqrt{8 k + 1} - 3} - 3} - 1}{2}

g (g (g (g (k)))) = \frac{\sqrt{4 \sqrt{4 \sqrt{4 \sqrt{8 k + 1} - 3} - 3} - 3} - 1}{2}

g (g (g (g (\frac{1}{32})))) = \frac{\sqrt{4 \sqrt{4 \sqrt{2 \sqrt{5} - 3} - 3} - 3} - 1}{2} \approx . 281886524469

Commented by mathdanisur last updated on 20/Sep/21

awesome solution thanks Ser