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Question Number 17595 by virus last updated on 08/Jul/17

$$Leta,b,cbetheposetiveintegersuchthat$$$$b/aisanalsointegerifa,b,careinGPand$$$$AMofa,b,cis(b+2)thenfindthevalueof$$$$(a^2+a-14)/(a+1)$$

Commented by virus last updated on 08/Jul/17

$$fine$$

Answered by mrW1 last updated on 08/Jul/17

$$\mathrm{b}=\mathrm{na}$$$$\frac{\mathrm{b}}{\mathrm{a}}=\frac{\mathrm{c}}{\mathrm{b}}$$$$\Rightarrow \mathrm{c}={\mathrm{n}}^2\mathrm{a}$$$$\frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{3}=\mathrm{b}+2$$$$\Rightarrow \frac{\mathrm{a}+\mathrm{na}+{\mathrm{n}}^2\mathrm{a}}{3}=\mathrm{na}+2$$$$\Rightarrow \mathrm{a}+\mathrm{na}+{\mathrm{n}}^2\mathrm{a}=3\mathrm{na}+6$$$$\Rightarrow {\mathrm{n}}^2-2\mathrm{n}+(1-\frac{6}{\mathrm{a}})=0$$$$\mathrm{n}=\frac{2+\sqrt{4-4(1-\frac{6}{\mathrm{a}})}}{2}=1+\sqrt{\frac{6}{\mathrm{a}}}$$$$\Rightarrow \mathrm{a}=6$$$$\Rightarrow \mathrm{n}=2\Rightarrow \mathrm{b}=12\Rightarrow \mathrm{c}=24$$$$({\mathrm{a}}^2+\mathrm{a}-14)/(\mathrm{a}+1)=(36+6-14)/7=4$$

Commented by virus last updated on 08/Jul/17

$$thankyousir$$

Commented by chux last updated on 09/Jul/17

$$\mathrm{mr}\mathrm{W}1....\mathrm{please}\mathrm{how}\mathrm{is}\mathrm{n}=2$$

Commented by mrW1 last updated on 09/Jul/17

$$\mathrm{n}=1+\sqrt{\frac{6}{\mathrm{a}}}$$$$\mathrm{since}\mathrm{n}\mathrm{and}\mathrm{a}\mathrm{are}\mathrm{positive}\mathrm{integers},$$$$\mathrm{the}\mathrm{only}\mathrm{solution}\mathrm{is}\mathrm{a}=6\mathrm{and}\mathrm{then}$$$$\mathrm{n}=2.$$

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