Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

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

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

$${leg}\:{A}_{\mathrm{1}} ,{A}_{\mathrm{2}} ,...{A}_{{n}} \:{and}\:{H}_{\mathrm{1}} ,{H}_{\mathrm{2}} ,...{H}_{{n}} \:{are}\:{n}\:{A}.{M}'{S}\: \\ $$$${and}\:{H}.{M}'{S}\:{respectively}\:{between}\:{a}\:{and}\:{b} \\ $$$${prove}\:{that}\:{A}_{{r}} {H}_{{n}−{r}+\mathrm{1}} ={ab} \\ $$$$\:{n}\geqslant{r}\geqslant\mathrm{1} \\ $$

Answered by Kunal12588 last updated on 20/Apr/19

a,A_1 ,A_2 ,...A_n ,b  are in AP  where  T_1 =a,  T_(n+2) =b  ⇒T_1 +(n+1)d=b  ⇒d=((b−a)/(n+1))  A_r =T_1 +rd  ⇒A_r =a+r(((b−a)/(n+1)))  ⇒A_r =((an+a+rb−ra)/(n+1))  ⇒A_r =((a(n−r+1)+rb)/(n+1))  a,H_1 ,H_2 ,...H_n ,b  are in HP  ⇒(1/a),(1/H_1 ),(1/H_2 ),...,(1/H_n ),(1/b) are in AP  ⇒(1/H_r )=(((1/a)(n−r+1)+r(1/b))/(n+1))=((b(n−r+1)+ra)/(ab(n+1)))  ⇒H_r =((ab(n+1))/(b(n−r+1)+ra))  H_(n−r+1) =((ab(n+1))/(b(n−(n−r+1)+1)+a(n−r+1)))  ⇒H_(n−r+1) =((ab(n+1))/(br+a(n−r+1)))  ∴A_r H_(n−r+1) =((a(n−r+1)+rb)/(n+1))×((ab(n+1))/(br+a(n−r+1)))  ⇒A_r H_(n−r+1) =ab(1)  ⇒A_r H_(n−r+1) =ab

$${a},{A}_{\mathrm{1}} ,{A}_{\mathrm{2}} ,...{A}_{{n}} ,{b} \\ $$$${are}\:{in}\:{AP} \\ $$$${where} \\ $$$${T}_{\mathrm{1}} ={a}, \\ $$$${T}_{{n}+\mathrm{2}} ={b} \\ $$$$\Rightarrow{T}_{\mathrm{1}} +\left({n}+\mathrm{1}\right){d}={b} \\ $$$$\Rightarrow{d}=\frac{{b}−{a}}{{n}+\mathrm{1}} \\ $$$${A}_{{r}} ={T}_{\mathrm{1}} +{rd} \\ $$$$\Rightarrow{A}_{{r}} ={a}+{r}\left(\frac{{b}−{a}}{{n}+\mathrm{1}}\right) \\ $$$$\Rightarrow{A}_{{r}} =\frac{{an}+{a}+{rb}−{ra}}{{n}+\mathrm{1}} \\ $$$$\Rightarrow{A}_{{r}} =\frac{{a}\left({n}−{r}+\mathrm{1}\right)+{rb}}{{n}+\mathrm{1}} \\ $$$${a},{H}_{\mathrm{1}} ,{H}_{\mathrm{2}} ,...{H}_{{n}} ,{b} \\ $$$${are}\:{in}\:{HP} \\ $$$$\Rightarrow\frac{\mathrm{1}}{{a}},\frac{\mathrm{1}}{{H}_{\mathrm{1}} },\frac{\mathrm{1}}{{H}_{\mathrm{2}} },...,\frac{\mathrm{1}}{{H}_{{n}} },\frac{\mathrm{1}}{{b}}\:{are}\:{in}\:{AP} \\ $$$$\Rightarrow\frac{\mathrm{1}}{{H}_{{r}} }=\frac{\frac{\mathrm{1}}{{a}}\left({n}−{r}+\mathrm{1}\right)+{r}\frac{\mathrm{1}}{{b}}}{{n}+\mathrm{1}}=\frac{{b}\left({n}−{r}+\mathrm{1}\right)+{ra}}{{ab}\left({n}+\mathrm{1}\right)} \\ $$$$\Rightarrow{H}_{{r}} =\frac{{ab}\left({n}+\mathrm{1}\right)}{{b}\left({n}−{r}+\mathrm{1}\right)+{ra}} \\ $$$${H}_{{n}−{r}+\mathrm{1}} =\frac{{ab}\left({n}+\mathrm{1}\right)}{{b}\left({n}−\left({n}−{r}+\mathrm{1}\right)+\mathrm{1}\right)+{a}\left({n}−{r}+\mathrm{1}\right)} \\ $$$$\Rightarrow{H}_{{n}−{r}+\mathrm{1}} =\frac{{ab}\left({n}+\mathrm{1}\right)}{{br}+{a}\left({n}−{r}+\mathrm{1}\right)} \\ $$$$\therefore{A}_{{r}} {H}_{{n}−{r}+\mathrm{1}} =\frac{{a}\left({n}−{r}+\mathrm{1}\right)+{rb}}{{n}+\mathrm{1}}×\frac{{ab}\left({n}+\mathrm{1}\right)}{{br}+{a}\left({n}−{r}+\mathrm{1}\right)} \\ $$$$\Rightarrow{A}_{{r}} {H}_{{n}−{r}+\mathrm{1}} ={ab}\left(\mathrm{1}\right) \\ $$$$\Rightarrow{A}_{{r}} {H}_{{n}−{r}+\mathrm{1}} ={ab} \\ $$

Commented by tanmay last updated on 20/Apr/19

excellent...i have posted these question[from  old book of BSTAT questions...

$${excellent}...{i}\:{have}\:{posted}\:{these}\:{question}\left[{from}\right. \\ $$$${old}\:{book}\:{of}\:{BSTAT}\:{questions}... \\ $$

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

so far i know in todays scenario plenty of  engineers and doctors are there...but they are  not satisfied whatever they get in return...  you may see youtube video the hidden meaning  of company package offer to IIT pass out..  so listen to your inner one voice what he want..  i am 49 year old govt employee my %age  was never high but i was good in math.. what  i am today blessing of math...  so i think..1)you may try NDA(national defence academy)  exam based on 10+2 phy+chem+math  2)Appear in BSTAT(Bachelor of statictics) exam  on Math only  3)Target IAS exam and get training in Delhi  every body are not number one...  but they should judge themself ..what is fit  for them...  finally buy a book on career consultation or  listen experts opinion on youtube.

$${so}\:{far}\:{i}\:{know}\:{in}\:{todays}\:{scenario}\:{plenty}\:{of} \\ $$$${engineers}\:{and}\:{doctors}\:{are}\:{there}...{but}\:{they}\:{are} \\ $$$${not}\:{satisfied}\:{whatever}\:{they}\:{get}\:{in}\:{return}... \\ $$$${you}\:{may}\:{see}\:{youtube}\:{video}\:{the}\:{hidden}\:{meaning} \\ $$$${of}\:{company}\:{package}\:{offer}\:{to}\:{IIT}\:{pass}\:{out}.. \\ $$$${so}\:{listen}\:{to}\:{your}\:{inner}\:{one}\:{voice}\:{what}\:{he}\:{want}.. \\ $$$${i}\:{am}\:\mathrm{49}\:{year}\:{old}\:{govt}\:{employee}\:{my}\:\%{age} \\ $$$${was}\:{never}\:{high}\:{but}\:{i}\:{was}\:{good}\:{in}\:{math}..\:{what} \\ $$$${i}\:{am}\:{today}\:{blessing}\:{of}\:{math}... \\ $$$$\left.{so}\:{i}\:{think}..\mathrm{1}\right){you}\:{may}\:{try}\:{NDA}\left({national}\:{defence}\:{academy}\right) \\ $$$${exam}\:{based}\:{on}\:\mathrm{10}+\mathrm{2}\:{phy}+{chem}+{math} \\ $$$$\left.\mathrm{2}\right){Appear}\:{in}\:{BSTAT}\left({Bachelor}\:{of}\:{statictics}\right)\:{exam} \\ $$$${on}\:{Math}\:{only} \\ $$$$\left.\mathrm{3}\right){Target}\:{IAS}\:{exam}\:{and}\:{get}\:{training}\:{in}\:{Delhi} \\ $$$${every}\:{body}\:{are}\:{not}\:{number}\:{one}... \\ $$$${but}\:{they}\:{should}\:{judge}\:{themself}\:..{what}\:{is}\:{fit} \\ $$$${for}\:{them}... \\ $$$${finally}\:{buy}\:{a}\:{book}\:{on}\:{career}\:{consultation}\:{or} \\ $$$${listen}\:{experts}\:{opinion}\:{on}\:{youtube}. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com