Question

The first terms of an A.P and G.P are each 2/3. Their common and difference and

common ratio are each X and the sum of their first three terms are also equal.

Find the two possible values of X

Answer 1

$$a=b=\dfrac{2}{3}$$

$$a+a+x+a+2x=b+bx+bx^2$$

$$3(\dfrac{2}{3})+3x=\dfrac{2}{3}(1+x+x^2)$$

$$6+9x=2+2x+2x^2$$

$$2x^2-7x-4=0$$

$$D=(-7)^2-4(2)(-4)=81$$

$$x=\dfrac{7\pm \sqrt{81}}{2(2)}$$

$$x_1=\dfrac{7- 9}{4}=-\dfrac{1}{2}$$

$$x_2=\dfrac{7+ 9}{4}=4$$

$x\in\{-\dfrac{1}{2}, 4\}$