Question

Suppose a, b, c are distinct positive real numbers such that a, 2b, 3c are in a.P. And a, b, c are in g.P. The common ratio of g.P. Is

Answer 1

common ratio of G.P is 1,or 1/3

Step-by-step explanation:

Given,

a,b,c are in G.P and  $a,2b,3c$     are in A.P

find out common ratio of G.P

for G.P

$b/a=c/b$   ( Common ratio is same)                     (1)

$b^{2}=ac$

for A.P

$2b-a=3c-a$           ( common difference is same)

$4b=a+3c$

$4=a/b+3c/b$          (divide by b both side)

$4=b/c+3c/b$           replace a/b=b/c) from equation 1

$4=(b^{2}+3c^{2})/bc$

$4bc=b^{2}+3c^{2}$

$b^{2}-4bc+3c^{2}=0$

$b^{2}-3bc-bc+3c^{2}=0$

$b\left ( b-3c \right )-c\left ( b-3c \right )=0$

$\left ( b-c \right )\left ( b-3c \right )=0$

$b-c=0$

$b=c$

$c/b=1$

or

$b-3c=0$

$b=3c$

$c/b=1/3$

common ratio of G.P is 1 or 1/3