Q If a, b, c are distinct positive real numbers in G P and logca, logbc, logab are - Maths - Sequences and Series

Question

Q If a, b, c are distinct positive real numbers in G P and
logca, logbc, logab are in A P
then find the common difference of this A P

Aarushi Mishra · Accepted Answer

Given a,b and c are in G P
Let common difference of this G P
 be rb=arc=ar2logxb=logxar=logxa +logxrlogxc=logxar2=logxa+2logxrGiven logca, logbc and logab are in A
P
 Let common difference be d2 logbc=logca+logab2logxclogxb=logxalogxc+logxblogxa2logxc2logxa=logxa2logxb+logxb2logxc2logxa +2logxr2logxa=logxa2logxa +logxr+logxa +logxr2logxa +2logxrDivide by logxr32logxalogxr +22logxalogxr=logxalogxr2logxalogxr +1+logxalogxr +12logxalogxr +2Let t=logxalogxr2t +22t=t2t +1+t +12t +2t +22t-t +12t +2=t2t +1-t +22tt+2t +2t-t +12=ttt +1-t +22t+2t2 +2t-t2 -2t-1=tt2+t-t2 -4t-4-t+2=-t 3t+4t+2=t 3t+43t2+4t-t-2=03t2+3t-2=0Using quadritic formulax=-b±b2-4ac2at=-3±9+246=-3±336logxalogxr=logra=-3±3361logar=-3±336logar=6-3±33r=a6-3±33

Q. If a, b, c are distinct positive real numbers in G.P. and logca, logbc, logab are in A.P. then find the common difference of this A.P.

Q. If a, b, c are distinct positive real numbers in G.P. and log_ca, log_bc, log_ab are in A.P. then find the common difference of this A.P.