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If-f-x-is-a-function-defined-for-all-real-numbers-such-that-f-2x-x-f-x-and-f-1-2-find-the-value-of-f-64-Ans-65-

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Question Number 131332 by ZiYangLee last updated on 03/Feb/21
If f(x) is a function defined for all real numbers such that f(2x)=x+f(x) and f(1)=2, find the value of f(64). [Ans: 65]
$$\mathrm{If}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{function}\:\mathrm{defined}\:\mathrm{for}\:\mathrm{all}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$\mathrm{such}\:\mathrm{that}\:{f}\left(\mathrm{2}{x}\right)={x}+{f}\left({x}\right)\:\mathrm{and}\:{f}\left(\mathrm{1}\right)=\mathrm{2}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{f}\left(\mathrm{64}\right). \\ $$$$\left[\mathrm{Ans}:\:\mathrm{65}\right] \\ $$
Answered by mr W last updated on 03/Feb/21
f(2)=1+f(1)=1+2=3 f(4)=2+f(2)=2+3=5 f(8)=4+f(4)=4+5=9 f(16)=8+f(8)=8+9=17 f(32)=16+f(16)=16+17=33 f(64)=32+f(32)=32+33=65
$${f}\left(\mathrm{2}\right)=\mathrm{1}+{f}\left(\mathrm{1}\right)=\mathrm{1}+\mathrm{2}=\mathrm{3} \\ $$$${f}\left(\mathrm{4}\right)=\mathrm{2}+{f}\left(\mathrm{2}\right)=\mathrm{2}+\mathrm{3}=\mathrm{5} \\ $$$${f}\left(\mathrm{8}\right)=\mathrm{4}+{f}\left(\mathrm{4}\right)=\mathrm{4}+\mathrm{5}=\mathrm{9} \\ $$$${f}\left(\mathrm{16}\right)=\mathrm{8}+{f}\left(\mathrm{8}\right)=\mathrm{8}+\mathrm{9}=\mathrm{17} \\ $$$${f}\left(\mathrm{32}\right)=\mathrm{16}+{f}\left(\mathrm{16}\right)=\mathrm{16}+\mathrm{17}=\mathrm{33} \\ $$$${f}\left(\mathrm{64}\right)=\mathrm{32}+{f}\left(\mathrm{32}\right)=\mathrm{32}+\mathrm{33}=\mathrm{65} \\ $$

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