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Question Number 191790 by mathlove last updated on 30/Apr/23

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

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

Commented by BaliramKumar last updated on 01/May/23

if x=1 then all term (−ve) but ((103)/(102)) is (+ve)

$$\mathrm{if}\:\mathrm{x}=\mathrm{1}\:\mathrm{then}\:\mathrm{all}\:\mathrm{term}\:\left(−\mathrm{ve}\right)\:\mathrm{but}\:\frac{\mathrm{103}}{\mathrm{102}}\:\mathrm{is}\:\left(+\mathrm{ve}\right)\: \\ $$

Commented by mathlove last updated on 01/May/23

waht is sulotion

$${waht}\:{is}\:{sulotion} \\ $$

Commented by mathlove last updated on 01/May/23

Q type is right

$${Q}\:{type}\:{is}\:{right} \\ $$

Commented by mehdee42 last updated on 01/May/23

please check the correctness of the givrn statment again

$${please}\:{check}\:{the}\:{correctness}\:{of}\:{the}\:{givrn}\:{statment}\:{again} \\ $$

Commented by mr W last updated on 01/May/23

it is shown that the question is wrong! just put x=0 or x=1! you can not prove something which is not true!

$${it}\:{is}\:{shown}\:{that}\:{the}\:{question}\:{is}\:{wrong}! \\ $$$${just}\:{put}\:{x}=\mathrm{0}\:{or}\:{x}=\mathrm{1}! \\ $$$${you}\:{can}\:{not}\:{prove}\:{something}\:{which} \\ $$$${is}\:{not}\:{true}! \\ $$

Commented by BaliramKumar last updated on 01/May/23

you mean find the value of x ?

$$\mathrm{you}\:\mathrm{mean}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:{x}\:? \\ $$

Commented by mathlove last updated on 02/May/23

yes

$${yes} \\ $$

Answered by BaliramKumar last updated on 01/May/23

((2x)/(2×3×4)) − (4/(2×3×4)) + ((3x)/(3×4×5)) − (5/(3×4×5)) + ...........+ ((100x)/(100×101×102)) − ((102)/(100×101×102)) = ((103)/(102)) (x/(3×4)) − (1/(2×3)) + (x/(4×5)) − (1/(3×4)) + ...........+ (x/(101×102)) − (1/(100×101)) = ((103)/(102)) (x/(3×4)) − (1/(2×3)) + (x/(4×5)) − (1/(3×4)) + ...........+ (x/(101×102)) − (1/(100×101)) = ((103)/(102)) x[(1/3) − (1/(102))] −[(1/2) − (1/(101))] = ((103)/(102)) x[ ((34−1)/(102))] −[((101−2)/(2×101))] = ((103)/(102)) x[ ((33)/(102))] −[((99)/(202))] = ((103)/(102)) x[ ((11)/(34))] = ((103)/(102)) + ((99)/(202)) ((11x)/(34)) = ((7726)/(5151)) x = ((15452)/(3333))

$$\frac{\mathrm{2}{x}}{\mathrm{2}×\mathrm{3}×\mathrm{4}}\:−\:\frac{\mathrm{4}}{\mathrm{2}×\mathrm{3}×\mathrm{4}}\:+\:\frac{\mathrm{3}{x}}{\mathrm{3}×\mathrm{4}×\mathrm{5}}\:−\:\frac{\mathrm{5}}{\mathrm{3}×\mathrm{4}×\mathrm{5}}\:+\:...........+\:\frac{\mathrm{100}{x}}{\mathrm{100}×\mathrm{101}×\mathrm{102}}\:−\:\frac{\mathrm{102}}{\mathrm{100}×\mathrm{101}×\mathrm{102}}\:=\:\frac{\mathrm{103}}{\mathrm{102}}\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\frac{{x}}{\mathrm{3}×\mathrm{4}}\:−\:\frac{\mathrm{1}}{\mathrm{2}×\mathrm{3}}\:+\:\frac{{x}}{\mathrm{4}×\mathrm{5}}\:−\:\frac{\mathrm{1}}{\mathrm{3}×\mathrm{4}}\:+\:...........+\:\frac{{x}}{\mathrm{101}×\mathrm{102}}\:−\:\frac{\mathrm{1}}{\mathrm{100}×\mathrm{101}}\:=\:\frac{\mathrm{103}}{\mathrm{102}} \\ $$$$\frac{{x}}{\mathrm{3}×\mathrm{4}}\:−\:\frac{\mathrm{1}}{\mathrm{2}×\mathrm{3}}\:+\:\frac{{x}}{\mathrm{4}×\mathrm{5}}\:−\:\frac{\mathrm{1}}{\mathrm{3}×\mathrm{4}}\:+\:...........+\:\frac{{x}}{\mathrm{101}×\mathrm{102}}\:−\:\frac{\mathrm{1}}{\mathrm{100}×\mathrm{101}}\:=\:\frac{\mathrm{103}}{\mathrm{102}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$${x}\left[\frac{\mathrm{1}}{\mathrm{3}}\:−\:\frac{\mathrm{1}}{\mathrm{102}}\right]\:−\left[\frac{\mathrm{1}}{\mathrm{2}}\:−\:\frac{\mathrm{1}}{\mathrm{101}}\right]\:=\:\frac{\mathrm{103}}{\mathrm{102}} \\ $$$${x}\left[\:\frac{\mathrm{34}−\mathrm{1}}{\mathrm{102}}\right]\:−\left[\frac{\mathrm{101}−\mathrm{2}}{\mathrm{2}×\mathrm{101}}\right]\:=\:\frac{\mathrm{103}}{\mathrm{102}} \\ $$$${x}\left[\:\frac{\mathrm{33}}{\mathrm{102}}\right]\:−\left[\frac{\mathrm{99}}{\mathrm{202}}\right]\:=\:\frac{\mathrm{103}}{\mathrm{102}} \\ $$$${x}\left[\:\frac{\mathrm{11}}{\mathrm{34}}\right]\:\:=\:\frac{\mathrm{103}}{\mathrm{102}}\:+\:\frac{\mathrm{99}}{\mathrm{202}} \\ $$$$\frac{\mathrm{11}{x}}{\mathrm{34}}\:=\:\frac{\mathrm{7726}}{\mathrm{5151}} \\ $$$${x}\:=\:\frac{\mathrm{15452}}{\mathrm{3333}} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\: \\ $$

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