What is the common ratio of the geometric sequence 4, 2, 1, 0.5, 0.25, 0.125, ... ?

Introduction / Context:
This question checks knowledge of geometric sequences, specifically the concept of the common ratio. In a geometric sequence, each term is obtained by multiplying the previous term by a constant factor called the common ratio. Recognizing this ratio is essential for working with geometric progressions in many aptitude and mathematics problems. The given sequence is 4, 2, 1, 0.5, 0.25, 0.125, and so on.

Given Data / Assumptions:

• The sequence is 4, 2, 1, 0.5, 0.25, 0.125, ...
• The terms are in a geometric progression.
• We must find the common ratio r.
• The common ratio is constant between any consecutive pair of terms.

Concept / Approach:
For a geometric sequence, the common ratio r can be found by dividing any term by its previous term, provided we choose consecutive nonzero terms. That is, r = T(n+1) / T(n). In this sequence, we can use 2 / 4, or 1 / 2, or 0.5 / 1, etc. All these ratios should be equal. By evaluating this ratio and confirming it across several consecutive term pairs, we can identify the correct common ratio.

Step-by-Step Solution:
Step 1: Consider the first two terms, 4 and 2. Step 2: Compute the ratio of the second term to the first term: r = 2 / 4 = 0.5. Step 3: Check this ratio with the next pair of terms, 2 and 1. Step 4: Compute 1 / 2 = 0.5, which matches the earlier value. Step 5: Check one more pair, 1 and 0.5. Step 6: Compute 0.5 / 1 = 0.5, again the same value. Step 7: For 0.5 and 0.25, compute 0.25 / 0.5 = 0.5. Step 8: Since the ratio is consistently 0.5, the common ratio of this geometric sequence is 0.5.

Verification / Alternative check:
Starting from the first term 4 and using r = 0.5, the terms are generated as follows: second term = 4 * 0.5 = 2, third term = 2 * 0.5 = 1, fourth term = 1 * 0.5 = 0.5, fifth term = 0.5 * 0.5 = 0.25, and sixth term = 0.25 * 0.5 = 0.125. These match the given sequence exactly, verifying that the common ratio is correctly identified as 0.5.

Why Other Options Are Wrong:
Option (b) −1: Would cause the signs of terms to alternate between positive and negative, which is not observed here.
Option (c) 1.5: This would make each term larger than the previous, whereas in the given sequence each term is smaller.
Option (d) −0.5: This would again alternate signs, which is not the case in the given sequence.

Common Pitfalls:
A common mistake is to subtract terms instead of dividing, which would lead to confusing this with an arithmetic progression. Another error is to divide in the wrong order (previous over next), which still yields the reciprocal but can cause confusion about the ratio. Always take the ratio as next term divided by previous term for geometric sequences.

Final Answer:
The common ratio of the sequence is 0.5.