Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 71666 by aliesam last updated on 18/Oct/19

f:z→z f(x+y)=f(x)+f(y)+3(4xy−1) ,f(1)=0 ∀x,y ∈z evaluate f(19)

$${f}:{z}\rightarrow{z} \\ $$$$ \\ $$$${f}\left({x}+{y}\right)={f}\left({x}\right)+{f}\left({y}\right)+\mathrm{3}\left(\mathrm{4}{xy}−\mathrm{1}\right) \\ $$$$ \\ $$$$,{f}\left(\mathrm{1}\right)=\mathrm{0} \\ $$$$ \\ $$$$\forall{x},{y}\:\in{z} \\ $$$${evaluate}\:{f}\left(\mathrm{19}\right) \\ $$

Commented by kaivan.ahmadi last updated on 18/Oct/19

f(2)=f(1+1)=9 f(3)=f(1+2)=0+9+21=30 f(6)=f(3+3)=30+30+135=195 f(4)=f(2+2)=9+9+45=63 f(10)=f(6+4)=195+63+285=543 f(9)=f(3+6)=30+195+213=438 f(19)=f(10+9)=543+438+1077=2058

$${f}\left(\mathrm{2}\right)={f}\left(\mathrm{1}+\mathrm{1}\right)=\mathrm{9} \\ $$$${f}\left(\mathrm{3}\right)={f}\left(\mathrm{1}+\mathrm{2}\right)=\mathrm{0}+\mathrm{9}+\mathrm{21}=\mathrm{30} \\ $$$${f}\left(\mathrm{6}\right)={f}\left(\mathrm{3}+\mathrm{3}\right)=\mathrm{30}+\mathrm{30}+\mathrm{135}=\mathrm{195} \\ $$$${f}\left(\mathrm{4}\right)={f}\left(\mathrm{2}+\mathrm{2}\right)=\mathrm{9}+\mathrm{9}+\mathrm{45}=\mathrm{63} \\ $$$${f}\left(\mathrm{10}\right)={f}\left(\mathrm{6}+\mathrm{4}\right)=\mathrm{195}+\mathrm{63}+\mathrm{285}=\mathrm{543} \\ $$$${f}\left(\mathrm{9}\right)={f}\left(\mathrm{3}+\mathrm{6}\right)=\mathrm{30}+\mathrm{195}+\mathrm{213}=\mathrm{438} \\ $$$${f}\left(\mathrm{19}\right)={f}\left(\mathrm{10}+\mathrm{9}\right)=\mathrm{543}+\mathrm{438}+\mathrm{1077}=\mathrm{2058} \\ $$$$ \\ $$$$ \\ $$

Answered by mind is power last updated on 18/Oct/19

f(x+1)=f(x)+3(4x−3)=f(x)+12x−3 ⇒f(x+1)−f(x)=12x−3 ∀x∈Z ⇒Σ_(x=1) ^(18) [f(x+1)−f(x)]=Σ_(x=1) ^(18) (12x−9) ⇒f(19)−f(1)=((12+12.18)/2).18−3.18 ⇒f(19)=9(12+216)−54=1998

$$\mathrm{f}\left(\mathrm{x}+\mathrm{1}\right)=\mathrm{f}\left(\mathrm{x}\right)+\mathrm{3}\left(\mathrm{4x}−\mathrm{3}\right)=\mathrm{f}\left(\mathrm{x}\right)+\mathrm{12x}−\mathrm{3} \\ $$$$\Rightarrow\mathrm{f}\left(\mathrm{x}+\mathrm{1}\right)−\mathrm{f}\left(\mathrm{x}\right)=\mathrm{12x}−\mathrm{3}\:\:\forall\mathrm{x}\in\mathbb{Z} \\ $$$$\Rightarrow\underset{\mathrm{x}=\mathrm{1}} {\overset{\mathrm{18}} {\sum}}\left[\mathrm{f}\left(\mathrm{x}+\mathrm{1}\right)−\mathrm{f}\left(\mathrm{x}\right)\right]=\underset{\mathrm{x}=\mathrm{1}} {\overset{\mathrm{18}} {\sum}}\left(\mathrm{12x}−\mathrm{9}\right) \\ $$$$\Rightarrow\mathrm{f}\left(\mathrm{19}\right)−\mathrm{f}\left(\mathrm{1}\right)=\frac{\mathrm{12}+\mathrm{12}.\mathrm{18}}{\mathrm{2}}.\mathrm{18}−\mathrm{3}.\mathrm{18} \\ $$$$\Rightarrow\mathrm{f}\left(\mathrm{19}\right)=\mathrm{9}\left(\mathrm{12}+\mathrm{216}\right)−\mathrm{54}=\mathrm{1998} \\ $$$$ \\ $$$$ \\ $$

Commented by aliesam last updated on 18/Oct/19

thank you sir nice sol

$${thank}\:{you}\:{sir}\:{nice}\:{sol} \\ $$

Commented by mind is power last updated on 18/Oct/19

y′re welcom

$$\mathrm{y}'\mathrm{re}\:\mathrm{welcom} \\ $$$$ \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com