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if-f-x-1-f-x-3-and-f-25-72-find-f-2-

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Question Number 157686 by MathSh last updated on 26/Oct/21
if f(x+1)-f(x)=3 and f(25)=72 find f(2) = ?
$$\mathrm{if}\:\:\:\mathrm{f}\left(\mathrm{x}+\mathrm{1}\right)-\mathrm{f}\left(\mathrm{x}\right)=\mathrm{3}\:\:\:\mathrm{and}\:\:\:\mathrm{f}\left(\mathrm{25}\right)=\mathrm{72} \\ $$$$\mathrm{find}\:\:\:\mathrm{f}\left(\mathrm{2}\right)\:=\:? \\ $$
Answered by tounghoungko last updated on 26/Oct/21
⇒f(x)=f(x+1)−3 ⇒f(25)=f(24)−3 ⇒f(24)=f(23)−3 ⋮ ⇒f(3)=f(2)−3 −−−−−−−−−−− ⇒f(25)=23×(−3)+f(2) ⇒f(2)=72+69=141
$$\Rightarrow{f}\left({x}\right)={f}\left({x}+\mathrm{1}\right)−\mathrm{3} \\ $$$$\Rightarrow{f}\left(\mathrm{25}\right)={f}\left(\mathrm{24}\right)−\mathrm{3} \\ $$$$\Rightarrow{f}\left(\mathrm{24}\right)={f}\left(\mathrm{23}\right)−\mathrm{3} \\ $$$$\:\:\:\:\:\vdots \\ $$$$\Rightarrow{f}\left(\mathrm{3}\right)={f}\left(\mathrm{2}\right)−\mathrm{3} \\ $$$$−−−−−−−−−−− \\ $$$$\Rightarrow{f}\left(\mathrm{25}\right)=\mathrm{23}×\left(−\mathrm{3}\right)+{f}\left(\mathrm{2}\right) \\ $$$$\Rightarrow{f}\left(\mathrm{2}\right)=\mathrm{72}+\mathrm{69}=\mathrm{141} \\ $$
Answered by mr W last updated on 26/Oct/21
f(x+1)=f(x)+3 f(x+2)=f(x+1)+3=f(x)+2×3 ... f(x+n)=f(x)+3n f(25)=f(2)+3(25−2) f(2)=f(25)−3×23=72−3×23=3
$${f}\left({x}+\mathrm{1}\right)={f}\left({x}\right)+\mathrm{3} \\ $$$${f}\left({x}+\mathrm{2}\right)={f}\left({x}+\mathrm{1}\right)+\mathrm{3}={f}\left({x}\right)+\mathrm{2}×\mathrm{3} \\ $$$$… \\ $$$${f}\left({x}+{n}\right)={f}\left({x}\right)+\mathrm{3}{n} \\ $$$$ \\ $$$${f}\left(\mathrm{25}\right)={f}\left(\mathrm{2}\right)+\mathrm{3}\left(\mathrm{25}−\mathrm{2}\right) \\ $$$${f}\left(\mathrm{2}\right)={f}\left(\mathrm{25}\right)−\mathrm{3}×\mathrm{23}=\mathrm{72}−\mathrm{3}×\mathrm{23}=\mathrm{3} \\ $$
Commented by MathSh last updated on 26/Oct/21
Very nice dear Ser, thank you so much
$$\mathrm{Very}\:\mathrm{nice}\:\mathrm{dear}\:\boldsymbol{\mathrm{S}}\mathrm{er},\:\mathrm{thank}\:\mathrm{you}\:\mathrm{so}\:\mathrm{much} \\ $$
Answered by qaz last updated on 26/Oct/21
f(x+1)−f(x)=3 Σ_(k=x) ^n (f(k+1)−f(k))=Σ_(k=x) ^n 3 ⇒f(n+1)−f(x)=3(n+1−x) f(25)−f(2)=3(25−2)=69 ⇒f(2)=72−69=3
$$\mathrm{f}\left(\mathrm{x}+\mathrm{1}\right)−\mathrm{f}\left(\mathrm{x}\right)=\mathrm{3} \\ $$$$\underset{\mathrm{k}=\mathrm{x}} {\overset{\mathrm{n}} {\sum}}\left(\mathrm{f}\left(\mathrm{k}+\mathrm{1}\right)−\mathrm{f}\left(\mathrm{k}\right)\right)=\underset{\mathrm{k}=\mathrm{x}} {\overset{\mathrm{n}} {\sum}}\mathrm{3} \\ $$$$\Rightarrow\mathrm{f}\left(\mathrm{n}+\mathrm{1}\right)−\mathrm{f}\left(\mathrm{x}\right)=\mathrm{3}\left(\mathrm{n}+\mathrm{1}−\mathrm{x}\right) \\ $$$$\mathrm{f}\left(\mathrm{25}\right)−\mathrm{f}\left(\mathrm{2}\right)=\mathrm{3}\left(\mathrm{25}−\mathrm{2}\right)=\mathrm{69} \\ $$$$\Rightarrow\mathrm{f}\left(\mathrm{2}\right)=\mathrm{72}−\mathrm{69}=\mathrm{3} \\ $$
Commented by MathSh last updated on 26/Oct/21
Very nice dear Ser, thank you so much
$$\mathrm{Very}\:\mathrm{nice}\:\mathrm{dear}\:\boldsymbol{\mathrm{S}}\mathrm{er},\:\mathrm{thank}\:\mathrm{you}\:\mathrm{so}\:\mathrm{much} \\ $$

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