If a, b, c, are in AP.
a2, b2, c2 are in HP
prove that either a=b=c or
a,b,-c/2 are in GP
-
UP 0 DOWN 0 0 1
1 Answers
1a,b,c are in A.P
b-a=c-b
a2,b2,c2 are in H.P
1/b2-1/a2=1/c2-1/b2
(a-b)(a+b)/a2b2=(b-c)(b+c)/b2c2
a+b/a2=b+c/c2 (since a-b=b-c)
b(1/a2-1/c2)+(1/a-1/c)=0
(1/a-1/c)[b(1/a+1/c)+1]=0
either 1/a-1/c=0 a=c=b
or b(1/a+1/c)+1=0
b(a+c)=-ac
b2=-ac/2 a,b,-c/2 are in GP