Question

First three of the four numbers are in the A . P ,and the last three is H . P. Prove that the first four are proportional.

Answer 1

Let the four number of be a , b c d
$\therefore$ 2b = a + c
as a , b c are A . P
c = 2bd /( b + d )
$\because$ b , c , d are in H . P
c( b + d ) = d (a + c) , by (1) and (2)
bc + cd = da + cd $\therefore$ bc = da
or $\frac{a}{b}$ = $\frac{c}{d}$ i. e. for number are proportional