A number series is given below with one missing term. Choose the correct alternative from the given options that will complete the series. 100, -20, 4, ?, 0.16, -0.032

Introduction / Context:
This item checks understanding of geometric type number series where each term is obtained by multiplying the previous term by a constant factor. In this case the factor is a negative fraction, which changes both the sign and magnitude of each term. Such questions train the learner to recognise multiplication patterns beyond simple integers.

Given Data / Assumptions:
- The series provided is: 100, -20, 4, ?, 0.16, -0.032. - The fourth term is missing and needs to be determined. - The pattern is expected to be consistent across all consecutive terms.

Concept / Approach:
Number series that alternate in sign and decrease in magnitude often follow a multiplication rule with a negative fraction. To solve, compute the ratio of consecutive terms. If this ratio is constant, the series is geometric. The same ratio is then applied to move forward or backward in the sequence to find the missing term.

Step-by-Step Solution:
Step 1: Compute the ratio from the first term to the second term: -20 / 100 = -0.2. Step 2: Compute the ratio from the second term to the third term: 4 / (-20) = -0.2. Step 3: Since the ratio is constant and equal to -0.2, the series is geometric with common ratio -0.2. Step 4: To get the missing fourth term, multiply the third term by the common ratio: 4 * (-0.2) = -0.8. Step 5: Verify using the remaining terms. Multiply -0.8 by -0.2 to get 0.16, then multiply 0.16 by -0.2 to get -0.032. This matches the given fifth and sixth terms.

Verification / Alternative check:
An alternative check is to start from the known later term and work backward. For example, -0.032 divided by -0.2 gives 0.16, and 0.16 divided by -0.2 gives -0.8, which confirms the missing value and validates the common ratio -0.2 in both directions.

Why Other Options Are Wrong:
- Option 0.8 has the wrong sign, and 0.8 * (-0.2) would not give 0.16, so it breaks the pattern. - Option 1 cannot be obtained by multiplying 4 by -0.2 and does not lead correctly to 0.16. - Option -1 does not produce 0.16 when multiplied by -0.2 and therefore does not maintain the constant ratio.

Common Pitfalls:
Learners sometimes ignore the sign of the ratio and treat the pattern as if it had a positive common ratio. Others may compute the ratio with rounding or inexact division, which can hide a clear fractional value such as -0.2. It is important to handle negative signs carefully and to test the pattern across more than one pair of terms.

Final Answer:
The missing term that completes the geometric pattern is -0.8, so the correct option is -0.8.