Question

Let x, y ,z be in A.P. and their sum is 15. If 1 , 3 , 9 are added to these three numbers respectively , then they form a G.P. find x , y and z.

Answer 1

Let three first numbers in an arithmetic progression are $x, x+d, x+2d.$

Then $x+x+d+x+2d=15$

$$\dfrac{x+d+3}{x+1}=\dfrac{x+2d+9}{x+d+3}$$

$$3x+3d=15=>d=5-x$$

$$(x+5-x+3)^2=(x+1)(x+2(5-x)+9)$$

$$(x+1)(19-x)=64$$

$$-x^2+18x+19=64$$

$$x^2-18x+45=0$$

$$(x-3)(x-15)=0$$

$$x_1=3, x_2=15$$ $$d_1=2, d_2=-10$$

$x=3, y=5, z=7$

or

$x=15, y=5,z=-5$