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Question Number 135689 by joki last updated on 15/Mar/21

$$\mathrm{0},....,....,\frac{\mathrm{5}}{\mathrm{4}},\frac{\mathrm{4}}{\mathrm{5}},.... \\$$$$\mathrm{how}\:\mathrm{to}\:\mathrm{solution}\:\mathrm{aritmatic} \\$$

Commented by liberty last updated on 15/Mar/21

$${AP}\:? \\$$

Commented by joki last updated on 15/Mar/21

$$\mathrm{common}\:\mathrm{difference}\:\mathrm{and}\:\mathrm{sum}\: \\$$$$\\$$

Commented by mr W last updated on 15/Mar/21

$${if}\:{it}\:{is}\:{an}\:{AP},\:{then}\:{the}\:{common} \\$$$${difference}\:{is}\:\frac{\mathrm{4}}{\mathrm{5}}−\frac{\mathrm{5}}{\mathrm{4}}=−\frac{\mathrm{9}}{\mathrm{20}} \\$$$${that}\:{means}\:{the}\:{AP}\:{is}\:{decreasing}! \\$$$${all}\:{terms}\:{before}\:{the}\:{term}\:\frac{\mathrm{4}}{\mathrm{5}}\:{must} \\$$$${be}\:{larger}\:{than}\:\frac{\mathrm{4}}{\mathrm{5}},\:{this}\:{is}\:{not}\:{the}\:{case}, \\$$$${since}\:{the}\:{first}\:{term}\:{is}\:\mathrm{0},\:{which}\:{is} \\$$$${smaller}\:{than}\:\frac{\mathrm{4}}{\mathrm{5}}.\:{therefore}\:{the} \\$$$${question}\:{is}\:{bad},\:{it}'{s}\:{not}\:{an}\:{AP},\:{but} \\$$$${what}\:{it}\:{is},\:{is}\:{not}\:{clear}. \\$$

Commented by MJS_new last updated on 15/Mar/21

$$\mathrm{assuming}\:\mathrm{3}\:\mathrm{points}\:\mathrm{given} \\$$$${P}_{\mathrm{1}} =\begin{pmatrix}{\mathrm{1}}\\{\mathrm{0}}\end{pmatrix}\:\:\:{P}_{\mathrm{2}} =\begin{pmatrix}{\mathrm{4}}\\{\mathrm{5}/\mathrm{4}}\end{pmatrix}\:\:\:{P}_{\mathrm{3}} =\begin{pmatrix}{\mathrm{5}}\\{\mathrm{4}/\mathrm{5}}\end{pmatrix} \\$$$${y}={ax}^{\mathrm{2}} +{bx}+{c} \\$$$$\left(\mathrm{1}\right)\:\:\mathrm{0}={a}+{b}+{c} \\$$$$\left(\mathrm{2}\right)\:\:\frac{\mathrm{5}}{\mathrm{4}}=\mathrm{16}{a}+\mathrm{4}{b}+{c} \\$$$$\left(\mathrm{3}\right)\:\:\frac{\mathrm{4}}{\mathrm{5}}=\mathrm{25}{a}+\mathrm{5}{b}+{c} \\$$$$\mathrm{solving}\:\mathrm{this}\:\mathrm{system}\:\mathrm{I}\:\mathrm{get} \\$$$${a}=−\frac{\mathrm{13}}{\mathrm{60}}\wedge{b}=\frac{\mathrm{3}}{\mathrm{2}}\wedge{c}=−\frac{\mathrm{77}}{\mathrm{60}} \\$$$${y}=−\frac{\mathrm{13}}{\mathrm{60}}{x}^{\mathrm{2}} +\frac{\mathrm{3}}{\mathrm{2}}{x}−\frac{\mathrm{77}}{\mathrm{60}} \\$$