Question

The 6th term of a geometric progression (GP) is 48. The first term is 1.5. What is the common ratio (CR)?

• A. 2.0
• B. 3.0
• C. 2.5
• D. 2.2

Answer 1

Answer:

A. 2.0

Understanding Geometric Progression (GP) Formulas

• A Geometric Progression is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r).
• The formula for the n-th term of a GP is defined as: T_n = a × r^(n-1).
• Here, a represents the first term, r represents the common ratio, and n is the position of the term in the sequence.

Step-by-Step Calculation

1. Identify given values:
   • First term (a) = 1.5
   • 6th term (T₆) = 48
   • Position (n) = 6
2. Substitute values into the formula:
   • 48 = 1.5 × r^(6-1)
   • 48 = 1.5 × r⁵
3. Solve for r:
   • r⁵ = 48 / 1.5
   • r⁵ = 32
   • To isolate r, take the fifth root of 32: r = ⁵√32
   • Since 2⁵ = 32, the common ratio r = 2.

Important Exam Tips for GP

• Always verify the formula application: Ensure (n-1) is used for the exponent, not n.
• For powers like ⁵√32 or ³√27, memorizing small integer powers (e.g., 2⁵=32, 3⁴=81) significantly speeds up calculation during competitive exams.
• If the common ratio is negative, pay attention to the sign alternation in the sequence terms.