Question

Which of the following is not in Geometric Progression?

• A. \(-2,\ -6,\ -18,\ \ldots\)
• B. \(64,\ -32,\ 16,\ \ldots\)
• C. \(3,\ 6,\ 12,\ \ldots\)
• D. \(5,\ 55,\ 555,\ \ldots\)

Hint:

For a Geometric Progression: \[ \frac{\text{Next Term}}{\text{Previous Term}} \] must remain constant throughout the sequence. If the ratio changes even once, the sequence is not a G.P.

Answer 1

The Correct Option is D

Concept: A sequence is said to be in Geometric Progression (G.P.) if the ratio of any term to its preceding term remains constant throughout the sequence. If: \[ a,\ ar,\ ar^2,\ ar^3,\ \ldots \] then: \[ \frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = r \] where \(r\) is called the common ratio.

Step 1: Checking Option (1).
Sequence: \[ -2,\ -6,\ -18,\ \ldots \] Calculate the ratios: \[ \frac{-6}{-2} = 3 \] \[ \frac{-18}{-6} = 3 \] Since the ratio is constant, this is a G.P.

Step 2: Checking Option (2).
Sequence: \[ 64,\ -32,\ 16,\ \ldots \] Calculate the ratios: \[ \frac{-32}{64} = -\frac{1}{2} \] \[ \frac{16}{-32} = -\frac{1}{2} \] The ratio is constant. Hence, this is a G.P.

Step 3: Checking Option (3).
Sequence: \[ 3,\ 6,\ 12,\ \ldots \] Calculate the ratios: \[ \frac{6}{3} = 2 \] \[ \frac{12}{6} = 2 \] Since the ratio remains constant, this is also a G.P.

Step 4: Checking Option (4).
Sequence: \[ 5,\ 55,\ 555,\ \ldots \] Calculate the ratios: \[ \frac{55}{5} = 11 \] \[ \frac{555}{55} \approx 10.09 \] Since: \[ 11 \ne 10.09 \] the common ratio is not constant. Therefore, this sequence is not a Geometric Progression. Final Conclusion: The sequence: \[ 5,\ 55,\ 555,\ \ldots \] is not in Geometric Progression. Hence, the correct answer is option (4).