a,b,c are sides of a triangle and are in GP if log a - log 2b, log 2b -log - Maths - Sequences and Series

Question

a,b,c are sides of a triangle and are in GP if log a - log 2b, log
2b -log 3c and log 3c - log a are in AP then

A)angle A must be obtuse B) angle B must be obtuse
C) C=90 degree D) none of these

pls give the full solution and not only the option

Neha Sethi · Accepted Answer

Dear student
Given:a,b,c are in GP⇒b2=ac   
1  If x,y,z are in GP ⇒y2=xzAlso  loga-log2b,log2b-log3c,log3c-loga are in A
P⇒2log2b-log3c=  loga-log2b+log3c-loga  If x,y,z are in AP ⇒2y=x+z⇒2log2b-2log3c=log3c-log2b⇒2log2b+log2b=log3c+2log3c⇒3log2b=3log3c⇒log2b=log3c⇒2b=3c⇒b=3c2⇒ b2=9c24   and From 1, 9c24=ac⇒c=4a9Now putting the value of c=4a9 in  b2=9c24⇒b2=94×16a281=4a29By cosine formula:cosA=b2+c2-a22bc=4a29+16a281-a22×2a3×4a9=36a2+16a2-81a28116a227=-2948i
e  cosA<0⇒A>90°Hence obtuse
Regards

a,b,c are sides of a triangle and are in GP. if log a - log 2b, log 2b -log 3c and log 3c - log a are in AP then A)angle A must be obtuse B) angle B must be obtuse C) C=90 degree D) none of these pls give the full solution and not only the option

a,b,c are sides of a triangle and are in GP. if log a - log 2b, log 2b -log 3c and log 3c - log a are in AP then

A)angle A must be obtuse B) angle B must be obtuse
C) C=90 degree D) none of these

pls give the full solution and not only the option