leg-A-1-A-2-A-n-and-H-1-H-2-H-n-are-n-A-M-S-and-H-M-S-respectively-between-a-and-b-prove-that-A-r-H-n-r-1-ab-n-r-1-

Question Number 58248 by tanmay last updated on 20/Apr/19

$$leg{A}_1,A_2,\dots A_{n}and{H}_1,H_2,\dots H_{n}arenA.M^{\prime }S$$$$andH.M^{\prime }Srespectivelybetweenaandb$$$$provethat{A}_r{H}_{n-r+1}=ab$$$$n⩾r⩾1$$

Answered by Kunal12588 last updated on 20/Apr/19

$$a,A_1,A_2,\dots A_n,b$$$$areinAP$$$$where$$$$T_1=a,$$$$T_{n+2}=b$$$$\Rightarrow T_1+(n+1)d=b$$$$\Rightarrow d=\frac{b-a}{n+1}$$$$A_r=T_1+rd$$$$\Rightarrow A_r=a+r\left(\frac{b-a}{n+1}\right)$$$$\Rightarrow A_r=\frac{an+a+rb-ra}{n+1}$$$$\Rightarrow A_r=\frac{a(n-r+1)+rb}{n+1}$$$$a,H_1,H_2,\dots H_n,b$$$$areinHP$$$$\Rightarrow \frac{1}{a},\frac{1}{H_1},\frac{1}{H_2},\dots ,\frac{1}{H_n},\frac{1}{b}areinAP$$$$\Rightarrow \frac{1}{H_r}=\frac{\frac{1}{a}(n-r+1)+r\frac{1}{b}}{n+1}=\frac{b(n-r+1)+ra}{ab(n+1)}$$$$\Rightarrow H_r=\frac{ab(n+1)}{b(n-r+1)+ra}$$$$H_{n-r+1}=\frac{ab(n+1)}{b(n-(n-r+1)+1)+a(n-r+1)}$$$$\Rightarrow H_{n-r+1}=\frac{ab(n+1)}{br+a(n-r+1)}$$$$\therefore A_r{H}_{n-r+1}=\frac{a(n-r+1)+rb}{n+1}\times \frac{ab(n+1)}{br+a(n-r+1)}$$$$\Rightarrow A_r{H}_{n-r+1}=ab\left(1\right)$$$$\Rightarrow A_r{H}_{n-r+1}=ab$$

Commented by tanmay last updated on 20/Apr/19

$$excellent\dots ihavepostedthesequestion[from$$$$oldbookofBSTATquestions\dots$$

Commented by Kunal12588 last updated on 20/Apr/19

Sir I am in class 12. I got 70.8% in class 11, I know it's not good. What should I do after 12th. I am ok in maths, not so good in physics and very bad in chemistry.

Commented by tanmay last updated on 20/Apr/19

$$sofariknowintodaysscenarioplentyof$$$$engineersanddoctorsarethere\dots buttheyare$$$$notsatisfiedwhatevertheygetinreturn\dots$$$$youmayseeyoutubevideothehiddenmeaning$$$$ofcompanypackageoffertoIITpassout..$$$$solistentoyourinneronevoicewhathewant..$$$$iam49yearoldgovtemployeemy\mathrm{\%}age$$$$wasneverhighbutiwasgoodinmath..what$$$$iamtodayblessingofmath\dots$$$$soithink..1)youmaytryNDA\left(nationaldefenceacademy\right)$$$$exambasedon10+2phy+chem+math$$$$2)AppearinBSTAT\left(Bachelorofstatictics\right)exam$$$$onMathonly$$$$3)TargetIASexamandgettraininginDelhi$$$$everybodyarenotnumberone\dots$$$$buttheyshouldjudgethemself..whatisfit$$$$forthem\dots$$$$finallybuyabookoncareerconsultationor$$$$listenexpertsopiniononyoutube.$$

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