Answer to Question #245707 in Algebra for luka

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

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

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

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

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

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

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

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

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

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

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