Question

If a, b, c are in A.P. and a, mb, c are in G.P then prove that a, m^2 b, c are
in H.P

Answer 1

if a, b, c are in ap then 2b=a+c
if a , mb , c  are in gp  then m²b²=ac
using these relation you can get the ans

Answer 2

Answer:

Since,   $m^{2} b = \frac{2ac}{a+c}$    ,   $a,m^{2} , c$ are in H.P

Step-by-step explanation:

Step 1:

As  $a,b,c$   are in A.P,  $\implies 2b = a + c$  

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

As  $a, mb, c$  are in G.P, $\implies m^{2} b^{2} = ac$

                                     $\implies m^{2} b. [\frac{a+c}{2} ] = ac$  ........(Putting value of $b$ from (i) )

                                     $\implies$ $m^{2} b = \frac{2ac}{a+c}$

Since,   $m^{2} b = \frac{2ac}{a+c}$    ,   $a,m^{2} , c$ are in H.P

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