Question Number 12889 by sin (x) last updated on 05/May/17
$${f}\left({x}−\mathrm{1}\right)=\frac{\mathrm{2}}{\mathrm{3}}+{f}\left({x}\right) \\ $$$${f}\left(\mathrm{0}\right)=\mathrm{36}−{f}\left(\mathrm{21}\right) \\ $$$$\Rightarrow{f}\left(\mathrm{0}\right)=? \\ $$
Commented by prakash jain last updated on 06/May/17
$${f}\left({x}\right)−{f}\left({x}−\mathrm{1}\right)=−\frac{\mathrm{2}}{\mathrm{3}} \\ $$$${f}\left(\mathrm{0}\right)=\mathrm{36}−{f}\left(\mathrm{21}\right)\:\:\:\:\:….\left(\mathrm{1}\right) \\ $$$${f}\left(\mathrm{1}\right)−{f}\left(\mathrm{0}\right)=−\frac{\mathrm{2}}{\mathrm{3}}\:\:\:\:\:\:\: \\ $$$$.. \\ $$$${f}\left(\mathrm{21}\right)−{f}\left(\mathrm{20}\right)=−\frac{\mathrm{2}}{\mathrm{3}}\:\:\:\:….\left(\mathrm{22}\right) \\ $$$${adding}\:\left(\mathrm{1}\right)\:{to}\:\left(\mathrm{22}\right) \\ $$$${f}\left(\mathrm{21}\right)=−\frac{\mathrm{2}}{\mathrm{3}}×\mathrm{21}+\mathrm{36}−{f}\left(\mathrm{21}\right) \\ $$$$\mathrm{2}{f}\left(\mathrm{21}\right)=\mathrm{22} \\ $$$${f}\left(\mathrm{21}\right)=\mathrm{11} \\ $$$${substituting}\:{in}\:\left(\mathrm{1}\right) \\ $$$${f}\left(\mathrm{0}\right)=\mathrm{36}−\mathrm{11} \\ $$$${f}\left(\mathrm{0}\right)=\mathrm{25} \\ $$
Answered by prakash jain last updated on 06/May/17
$${f}\left(\mathrm{0}\right)=\mathrm{25} \\ $$
Answered by Joel577 last updated on 07/May/17
$${f}\left({x}\right)\:−\:{f}\left({x}\:−\:\mathrm{1}\right)\:=\:−\frac{\mathrm{2}}{\mathrm{3}} \\ $$$$\mathrm{Let}\:{f}\left({x}\right)\:=\:{ax}\:+\:{b} \\ $$$$\:\:\:\:\:\:\:\:{f}\left({x}\:−\:\mathrm{1}\right)\:=\:{a}\left({x}\:−\:\mathrm{1}\right)\:+\:{b}\:=\:{ax}\:−\:{a}\:+\:{b}\: \\ $$$$\left({ax}\:+\:{b}\right)\:−\:{ax}\:+\:{a}\:−\:{b}\:=\:−\frac{\mathrm{2}}{\mathrm{3}} \\ $$$${a}\:=\:−\frac{\mathrm{2}}{\mathrm{3}} \\ $$$${f}\left(\mathrm{21}\right)\:+\:{f}\left(\mathrm{0}\right)\:=\:\mathrm{36} \\ $$$$\left(−\frac{\mathrm{2}}{\mathrm{3}}\:.\:\mathrm{21}\:+\:{b}\right)\:+\:\left(−\frac{\mathrm{2}}{\mathrm{3}}\:.\:\mathrm{0}\:+\:{b}\right)\:=\:\mathrm{36} \\ $$$$−\mathrm{14}\:+\:\mathrm{2}{b}\:=\:\mathrm{36} \\ $$$${b}\:=\:\frac{\mathrm{36}\:+\:\mathrm{14}}{\mathrm{2}}\:=\:\mathrm{25} \\ $$$${f}\left(\mathrm{0}\right)\:=\:−\frac{\mathrm{2}}{\mathrm{3}}\:.\:\mathrm{0}\:+\:\mathrm{25}\:=\:\mathrm{25} \\ $$