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Question Number 46922 by 786786AM last updated on 02/Nov/18

$$\mathrm{Thr}6\mathrm{th}\mathrm{term}\mathrm{of}\mathrm{an}\mathrm{AP}\mathrm{is}\mathrm{equal}\mathrm{to}2,\mathrm{the}$$$$\mathrm{value}\mathrm{of}\mathrm{the}\mathrm{common}\mathrm{difference}\mathrm{of}\mathrm{the}\mathrm{AP}.$$$$\mathrm{Which}\mathrm{makes}\mathrm{the}\mathrm{product}{a}_1{a}_4{a}_5\mathrm{least}$$$$\mathrm{is}\mathrm{given}\mathrm{by}$$

Answered by tanmay.chaudhury50@gmail.com last updated on 03/Nov/18

$$a+5d=2$$$$p=a(a+3d)(a+4d)$$$$p=(2-5d)(2-5d+3d)(2-5d+4d)$$$$p=(2-5d)(2-2d)(2-d)$$$$lnp=ln(2-5d)+ln(2-2d)+ln(2-d)$$$$\frac{1}{p}\times \frac{dp}{d\left(d\right)}=\frac{1}{2-5d}\times (-5)+\frac{1}{2-2d}\times (-2)+\frac{1}{2-d}\times (-1)$$$$Dp=p\times [\frac{-5}{2-5d}+\frac{-2}{2-2d}+\frac{-1}{2-d}]$$$$=-5(2-2d)(2-d)-2(2-5d)(2-d)-(2-5d)(2-2d)$$$$=-5(4-6d+2d^2)-2(4-12d+5d^2)-1(4-14d+10d^2)$$$$formax/minDp=0$$$$20-30d+10d^2+8-24d+10d^2+4-14d+10d^2$$$$=30d^2-68d+32$$$$15d^2-34d+16=0$$$$d=\frac{34\pm \sqrt{{34}^2-4\times 15\times 16}}{2\times 15}$$$$d=\frac{34\pm \sqrt{1156-960}}{30}$$$$d=\frac{34\pm 14}{30}(\frac{48}{30},\frac{20}{30})\to (\frac{8}{5},\frac{2}{3})$$$$Dp=30d^2-68d+16$$$$D^{2}p=60d-68$$$$valueof{D}^{2}patd=\frac{8}{5}is$$$$60\times \frac{8}{5}-68=96-68=28+ve$$$$valueat\frac{2}{3}is$$$$60\times \frac{2}{3}-68=40-68=-28-ve$$$$soatd=\frac{2}{3}theproductismaximumandd=\frac{8}{5}$$$$itisminimum...$$$$nowa+5d=2$$$$a+5\times \frac{8}{5}=2a=-6$$$$a_1\times a_4\times a_5$$$$(-6)\times (-6+3\times \frac{8}{5})(-6+4\times \frac{8}{5})$$$$=(-6)\times \left(\frac{-30+24}{5}\right)\left(\frac{-30+32}{5}\right)$$$$=(-6)\times \left(\frac{-6}{5}\right)\left(\frac{2}{5}\right)=\frac{72}{25}plscheck....$$$$$$$$$$$$$$$$$$

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