If x - 1, x + 2, and 3x are three consecutive terms of a geometric progression (G.P.) with… — JAMB UTME

If $x - 1$ , $x + 2$ , and $3x$ are three consecutive terms of a geometric progression (G.P.) with positive terms, what is the common ratio of the progression?

$2$ Answer
B
$4$
C
$3$
D
$\frac{1}{2}$

Answer

The common ratio of the progression is

2

For any three consecutive terms in a G.P., the square of the middle term equals the product of the first and third terms. Solving

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

gives

2x^2 - 7x - 4 = 0

, which yields

x = 4

for positive terms. Substituting

x = 4

gives the terms

3, 6, 12

, which have a common ratio of

6 \div 3 = 2

Step-by-Step Solution

Set up the condition for consecutive terms in a Geometric Progression.

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

For three consecutive terms

a, b, c

in G.P., the middle term squared equals the product of the outer terms (

b^2 = ac

Expand both sides and rearrange into a quadratic equation.

x^2 + 4x + 4 = 3x^2 - 3x \implies 2x^2 - 7x - 4 = 0

Expanding allows gathering all terms on one side to solve for

x

Factorize the quadratic equation to find

x

(2x + 1)(x - 4) = 0 \implies x = 4

(since terms are positive,

x > 1

The solution

x = -\frac{1}{2}

gives negative terms, so

x = 4

is chosen.

Find the consecutive terms and calculate the common ratio

r

Terms are

4 - 1 = 3

4 + 2 = 6

, and

3(4) = 12

. Common ratio

r = \frac{6}{3} = 2

Dividing the second term by the first term gives the common ratio

r

Key Concept

Geometric Progression Consecutive Terms Property (

b^2 = ac

)

Estimated Time:1m 30s

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