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Question Number 12827 by 786786AM last updated on 03/May/17

$$\mathrm{Sum}\mathrm{of}\mathrm{three}\mathrm{numbers}\mathrm{in}\mathrm{GP}\mathrm{be}14.\mathrm{If}\mathrm{one}\mathrm{is}$$$$\mathrm{added}\mathrm{to}\mathrm{first}\mathrm{and}\mathrm{second}\mathrm{and}1\mathrm{is}\mathrm{subtracted}$$$$\mathrm{from}\mathrm{the}\mathrm{third},\mathrm{the}\mathrm{new}\mathrm{numbers}\mathrm{are}\mathrm{in}\mathrm{AP}.$$$$\mathrm{The}\mathrm{smallest}\mathrm{of}\mathrm{them}\mathrm{is}$$

Answered by mrW1 last updated on 04/May/17

$$a_1=a$$$$a_2=aq$$$$a_3={aq}^2$$$$a_1+a_2+a_3=14$$$$a(1+q+q^2)=14$$$$$$$$(a_2+1)-(a_1+1)=(a_3-1)-(a_2+1)$$$$2(a_2+1)=(a_1+1)+(a_3-1)$$$$2(aq+1)=a+1+{aq}^2-1$$$${aq}^2-2aq+a-2=0$$$${aq}^2+aq+a-3aq-2=0$$$$14-3aq-2=0$$$$aq=4$$$$a+aq+{aq}^2=a+4+\frac{16}{a}=14$$$$a+\frac{16}{a}=10$$$$a^2-10a+16=0$$$$a=\frac{10\pm \sqrt{100-4\times 16}}{2}=\frac{10\pm 6}{2}=8or2$$$$q=\frac{1}{2}or2$$$$thenumbersare2,4,8or8,4,2$$$$$$$$\Rightarrow thesmallestofthemis2.$$

Answered by sandy_suhendra last updated on 04/May/17

$$\mathrm{the}\mathrm{other}\mathrm{way},\mathrm{we}\mathrm{can}\mathrm{start}\mathrm{from}\mathrm{AP}$$$$\mathrm{AP}:(\mathrm{a}-\mathrm{b}),\mathrm{a},(\mathrm{a}+\mathrm{b})$$$$\mathrm{GP}:(\mathrm{a}-\mathrm{b}-1),(\mathrm{a}-1),(\mathrm{a}+\mathrm{b}+1)$$$$(\mathrm{a}-\mathrm{b}-1)+(\mathrm{a}-1)+(\mathrm{a}+\mathrm{b}+1)=14$$$$3\mathrm{a}-1=14$$$$3\mathrm{a}=15\Rightarrow \mathrm{a}=5$$$$\mathrm{GP}:(4-\mathrm{b}),4,(6+\mathrm{b})$$$$(4-\mathrm{b})(6+\mathrm{b})=4^2$$$$24-2\mathrm{b}-{\mathrm{b}}^2=16$$$${\mathrm{b}}^2+2\mathrm{b}-8=0$$$$(\mathrm{b}+4)(\mathrm{b}-2)=0$$$$\mathrm{b}=-4\Rightarrow 8,4,2$$$$\mathrm{b}=2\Rightarrow 2,4,8$$

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