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Question Number 157686 by MathSh last updated on 26/Oct/21

$$\mathrm{if}\mathrm{f}(\mathrm{x}+1)-\mathrm{f}\left(\mathrm{x}\right)=3\mathrm{and}\mathrm{f}\left(25\right)=72$$$$\mathrm{find}\mathrm{f}\left(2\right)=?$$

Answered by tounghoungko last updated on 26/Oct/21

$$\Rightarrow f\left(x\right)=f(x+1)-3$$$$\Rightarrow f\left(25\right)=f\left(24\right)-3$$$$\Rightarrow f\left(24\right)=f\left(23\right)-3$$$$⋮$$$$\Rightarrow f\left(3\right)=f\left(2\right)-3$$$$-----------$$$$\Rightarrow f\left(25\right)=23\times (-3)+f\left(2\right)$$$$\Rightarrow f\left(2\right)=72+69=141$$

Answered by mr W last updated on 26/Oct/21

$$f(x+1)=f\left(x\right)+3$$$$f(x+2)=f(x+1)+3=f\left(x\right)+2\times 3$$$$...$$$$f(x+n)=f\left(x\right)+3n$$$$$$$$f\left(25\right)=f\left(2\right)+3(25-2)$$$$f\left(2\right)=f\left(25\right)-3\times 23=72-3\times 23=3$$

Commented by MathSh last updated on 26/Oct/21

$$\mathrm{Very}\mathrm{nice}\mathrm{dear}\mathbf{S}\mathrm{er},\mathrm{thank}\mathrm{you}\mathrm{so}\mathrm{much}$$

Answered by qaz last updated on 26/Oct/21

$$\mathrm{f}(\mathrm{x}+1)-\mathrm{f}\left(\mathrm{x}\right)=3$$$$\underset{\mathrm{k}=\mathrm{x}}{\sum ^{\mathrm{n}}}(\mathrm{f}(\mathrm{k}+1)-\mathrm{f}\left(\mathrm{k}\right))=\underset{\mathrm{k}=\mathrm{x}}{\sum ^{\mathrm{n}}}3$$$$\Rightarrow \mathrm{f}(\mathrm{n}+1)-\mathrm{f}\left(\mathrm{x}\right)=3(\mathrm{n}+1-\mathrm{x})$$$$\mathrm{f}\left(25\right)-\mathrm{f}\left(2\right)=3(25-2)=69$$$$\Rightarrow \mathrm{f}\left(2\right)=72-69=3$$

Commented by MathSh last updated on 26/Oct/21

$$\mathrm{Very}\mathrm{nice}\mathrm{dear}\mathbf{S}\mathrm{er},\mathrm{thank}\mathrm{you}\mathrm{so}\mathrm{much}$$

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