Question

If \(x-1,\ x+2\) and \(x+8\) are three consecutive terms of a Geometric Progression, then the value of \(x\) is:

• A. \(2\)
• B. \(3\)
• C. \(4\)
• D. \(5\)

Hint:

For any three consecutive terms: \[ a,\ b,\ c \] in a Geometric Progression: \[ b^2 = ac \] This shortcut helps solve G.P. problems very quickly.

Answer 1

The Correct Option is C

Concept: A Geometric Progression (G.P.) is a sequence in which the ratio between consecutive terms remains constant. If: \[ a,\ b,\ c \] are three consecutive terms of a G.P., then: \[ \frac{b}{a} = \frac{c}{b} \] Cross-multiplying: \[ b^2 = ac \] This is one of the most important properties of Geometric Progression. It states that:
• The square of the middle term equals the product of the first and third terms.
• The middle term is called the geometric mean of the other two terms.

Step 1: Identifying the three consecutive terms.
The given terms are: \[ x-1,\ x+2,\ x+8 \] Comparing with: \[ a,\ b,\ c \] we get: \[ a = x-1 \] \[ b = x+2 \] \[ c = x+8 \]

Step 2: Applying the Geometric Progression condition.
Using the G.P. property: \[ b^2 = ac \] Substituting the given expressions: \[ (x+2)^2 = (x-1)(x+8) \] This equation will help us determine the value of \(x\).

Step 3: Expanding the left-hand side carefully.
Using the identity: \[ (a+b)^2 = a^2 + 2ab + b^2 \] we get: \[ (x+2)^2 = x^2 + 2(x)(2) + 2^2 \] \[ = x^2 + 4x + 4 \] Thus: \[ (x+2)^2 = x^2 + 4x + 4 \]

Step 4: Expanding the right-hand side carefully.
Now expand: \[ (x-1)(x+8) \] Using distributive multiplication: \[ = x(x) + x(8) -1(x) -1(8) \] \[ = x^2 + 8x - x - 8 \] \[ = x^2 + 7x - 8 \] Thus: \[ (x-1)(x+8) = x^2 + 7x - 8 \]

Step 5: Equating both expanded expressions.
We have: \[ x^2 + 4x + 4 = x^2 + 7x - 8 \] Subtract \(x^2\) from both sides: \[ 4x + 4 = 7x - 8 \] Bring the \(x\)-terms to one side and constants to the other side: \[ 4 + 8 = 7x - 4x \] \[ 12 = 3x \] Divide both sides by \(3\): \[ x = \frac{12}{3} \] \[ x = 4 \]

Step 6: Verification of the obtained answer.
Substitute: \[ x = 4 \] The three terms become: \[ 4-1 = 3 \] \[ 4+2 = 6 \] \[ 4+8 = 12 \] Thus, the sequence is: \[ 3,\ 6,\ 12 \] Check the common ratios: \[ \frac{6}{3} = 2 \] \[ \frac{12}{6} = 2 \] Since both ratios are equal, the terms are indeed in Geometric Progression. Hence, the value obtained is correct.

Step 7: Checking the options carefully.
Option (1): \[ 2 \] Incorrect. Option (2): \[ 3 \] Incorrect. Option (3): \[ 4 \] Correct. Option (4): \[ 5 \] Incorrect. Final Conclusion: The value of \(x\) is: \[ \boxed{4} \] Hence, the correct answer is option (3).