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Question Number 191790 Answers: 1 Comments: 7

prove that ((2x−4)/(2∙3∙4))+((3x−5)/(3∙4∙5))+((4x−6)/(4∙5∙6))+.....+((100x−102)/(100∙101∙102))=((103)/(102))

$${prove}\:{that} \\ $$$$\frac{\mathrm{2}{x}−\mathrm{4}}{\mathrm{2}\centerdot\mathrm{3}\centerdot\mathrm{4}}+\frac{\mathrm{3}{x}−\mathrm{5}}{\mathrm{3}\centerdot\mathrm{4}\centerdot\mathrm{5}}+\frac{\mathrm{4}{x}−\mathrm{6}}{\mathrm{4}\centerdot\mathrm{5}\centerdot\mathrm{6}}+.....+\frac{\mathrm{100}{x}−\mathrm{102}}{\mathrm{100}\centerdot\mathrm{101}\centerdot\mathrm{102}}=\frac{\mathrm{103}}{\mathrm{102}} \\ $$$$ \\ $$

Question Number 191787 Answers: 0 Comments: 0

Ques. 2 (Metric Space Question) Let d be a metric on a non−empty set X. Show that the function U is defined by U(x,y)=((d(x,y))/(1+d(x,y))), where x and y are arbitrary element X is also a metric on X.

$$\mathrm{Ques}.\:\mathrm{2}\:\left(\mathrm{Metric}\:\mathrm{Space}\:\mathrm{Question}\right) \\ $$$$\:\:\:\:\:\:\mathrm{Let}\:\mathrm{d}\:\mathrm{be}\:\mathrm{a}\:\mathrm{metric}\:\mathrm{on}\:\mathrm{a}\:\mathrm{non}−\mathrm{empty} \\ $$$$\mathrm{set}\:\mathrm{X}.\:\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{function}\:\mathrm{U}\:\mathrm{is} \\ $$$$\mathrm{defined}\:\mathrm{by}\:\mathrm{U}\left(\mathrm{x},\mathrm{y}\right)=\frac{\mathrm{d}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{1}+\mathrm{d}\left(\mathrm{x},\mathrm{y}\right)},\:\mathrm{where} \\ $$$$\mathrm{x}\:\mathrm{and}\:\mathrm{y}\:\mathrm{are}\:\mathrm{arbitrary}\:\mathrm{element}\:\mathrm{X}\:\mathrm{is}\:\mathrm{also} \\ $$$$\mathrm{a}\:\mathrm{metric}\:\mathrm{on}\:\mathrm{X}. \\ $$

Question Number 191786 Answers: 0 Comments: 0

Ques. 1 (Metric Space Question) Let X = ρ_∞ be the set of all bounded sequences of complex numbers. That is every element of ρ_∞ is a complex sequence x^− ={x^− }_(k=1) ^∞ such ∣x_i ∣<Kx^− , i=1,2,3,... where Kx is a real number which may define on x for an arbitrary x^− ={x_i }_(i=1) ^∞ and y^− ={y_i }_(i=1) ^∞ in ρ_∞ we define as d_∞ (x,y)=Sup∣x_i −y_i ∣, Verify that d_∞ is a metric on ρ_(∞.)

$$\mathrm{Ques}.\:\mathrm{1}\:\left(\mathrm{Metric}\:\mathrm{Space}\:\mathrm{Question}\right) \\ $$$$\:\:\:\:\:\:\:\:\mathrm{Let}\:\mathrm{X}\:=\:\rho_{\infty} \:\mathrm{be}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{all}\: \\ $$$$\mathrm{bounded}\:\mathrm{sequences}\:\mathrm{of}\:\mathrm{complex}\: \\ $$$$\mathrm{numbers}.\:\mathrm{That}\:\mathrm{is}\:\mathrm{every}\:\mathrm{element}\:\mathrm{of} \\ $$$$\rho_{\infty} \:\mathrm{is}\:\mathrm{a}\:\mathrm{complex}\:\mathrm{sequence}\:\overset{−} {\mathrm{x}}=\left\{\overset{−} {\mathrm{x}}\right\}_{\mathrm{k}=\mathrm{1}} ^{\infty} \: \\ $$$$\mathrm{such}\:\mid\mathrm{x}_{\mathrm{i}} \mid<\mathrm{K}\overset{−} {\mathrm{x}},\:\mathrm{i}=\mathrm{1},\mathrm{2},\mathrm{3},...\:\mathrm{where}\:\mathrm{Kx} \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{real}\:\mathrm{number}\:\mathrm{which}\:\mathrm{may}\:\mathrm{define} \\ $$$$\mathrm{on}\:\mathrm{x}\:\mathrm{for}\:\mathrm{an}\:\mathrm{arbitrary}\:\overset{−} {\mathrm{x}}=\left\{\mathrm{x}_{\mathrm{i}} \right\}_{\mathrm{i}=\mathrm{1}} ^{\infty} \:\mathrm{and} \\ $$$$\overset{−} {\mathrm{y}}=\left\{\mathrm{y}_{\mathrm{i}} \right\}_{\mathrm{i}=\mathrm{1}} ^{\infty} \:\mathrm{in}\:\rho_{\infty} \mathrm{we}\:\mathrm{define}\:\mathrm{as} \\ $$$$\mathrm{d}_{\infty} \left(\mathrm{x},\mathrm{y}\right)=\mathrm{Sup}\mid\mathrm{x}_{\mathrm{i}} −\mathrm{y}_{\mathrm{i}} \mid,\:\mathrm{Verify}\:\mathrm{that} \\ $$$$\mathrm{d}_{\infty} \:\mathrm{is}\:\mathrm{a}\:\mathrm{metric}\:\mathrm{on}\:\rho_{\infty.} \\ $$

Question Number 191775 Answers: 2 Comments: 0