Math questionWhat value of x makes the three terms x, x/(x + 1) , 3x/(x + 1)(x + 2) those of a geometric sequence?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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The terms x, x/(x + 1) , 3x/(x + 1)(x + 2) form a geometric series.

(x/(x + 1))/x = [3x/(x + 1)(x + 2)]/[x/(x + 1)]

=> 1/(x + 1) = 3/(x + 2)

=> x + 2 = 3x + 3

=> 2x = -1

=> x= -1/2

The value of x = -1/2

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll use the theorem of the geometric mean of 3 consecutive terms:x, x/(x + 1) , 3x/(x + 1)(x + 2)

[x/(x + 1)]^2 = x*3x/(x + 1)(x + 2)

We'll raise to square to the left side:

x^2/(x+1)^2 = 3x^2/(x + 1)(x + 2)

We'll simplify by (x+1) both sides:

x^2/(x+1) = 3x^2/(x+2)

We'll simplify by x^2 both sides:

1/(x+1) = 3/(x+2)

We'll cross multiply and we'll get:

3x + 3 = x + 2

We'll subtract x + 2 both sides:

2x + 1 = 0

We'll subtract 1:

2x = -1

x = -1/2

For x = -1/2, the 3 given terms are the terms of a geometric progression.

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