CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Let a1,a2,... be positive real numbers in geometric progression . For each n, let An,Gn,Hn be respectively, the A.M, G.M and H.M of a1,a2,an. Find an expression for the G.M of G1,G2,...Gn in terms of A1,A2,...An and H1,H2,...Hn

A
Gn=(A2A4.....AnH1H2....H2n)1/2n
No worries! We‘ve got your back. Try BYJU‘S free classes today!
B
Gn=(A1A2.....AnH2H1....H2n)1/2n
No worries! We‘ve got your back. Try BYJU‘S free classes today!
C
Gn=(A1A2.....AnH1H2....Hn)1/2n
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
D
None of these
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution

The correct option is C Gn=(A1A2.....AnH1H2....Hn)1/2n
Let Gm be the geometric mean of G1,G2,...,Gn.
Gm=(G1.G2...Gn)1n
=[(a1).(a1.a1r)12.(a1.a1r.a1r2)13...(a1.a1r.a1r2...a1rn1)1n]1n
where r is the common ratio of GPa1,a2,...,an.
=[(a1.a1...ntimes)(r12.r33.r64...r(n1)n2n)]1n
=[an1.r12+1+32+...+n12]1n
=a1[r12[(n1)n2]]1n=a1[rn14] ...(1)
Now, An=a1+a2+...+ann=a1(1rn)n(1r)
and Hn=n(1a1+1a2+...+1an)=n1a1(1+1r+...+1rn1)=a1n(1r)rn11rn
An.Hn=a1(1rn)n(1r)×a1n(1r)rn1(1rn)=a21rn1
nk=1AkHk=nk=1(a21rn1)
=(a21.a21.a21...ntimes)×r0.r1.r2...rn1
=a2n1.r1+2+...+(n1)=a2n1rn(n1)2
=[a1rn14]2n=[Gm]2n [from Eq.(1)]
Gm=[nk=1AkHk]12n
Gm=(A1A2...AnH1H2...Hn)12n

flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Fear of the Dark
QUANTITATIVE APTITUDE
Watch in App
Join BYJU'S Learning Program
CrossIcon