Question

If a b c are in ap b c d are in gp and c d e are in hp them prove that a c e will be in gp

Answer 1

Please see the attachment

Answer 2

Step-by-step explanation:

We have been given that

a b c are in ap . Thus, we have

$b=\frac{a+c}{2}...(i)$

b c d are in gp. Thus, we have

$c^2=bd...(ii)$

c d e are in hp. Thus, we have

$d=\frac{2ec}{e+c}$

Substituting the value of d from (ii)

$\frac{c^2}{b}=\frac{2ec}{e+c}$

Cancel, c from both sides of the numerator

$\frac{c}{b}=\frac{2e}{e+c}$

Cross multiplying, we get

$ec+c^2=2be$

Substituting the value of b from (i)

$ec+c^2=2(\frac{a+b}{2})e\\\\ec+c^2=ae+ec$

Cancel ae both sides

$c^2=ae$

Thus, we can conclude that a, c, e are in GP.