A series is given with one number missing. Choose the correct alternative from the given options that will complete the series: 400, -200, 100, ?, 25, -12.5

Introduction / Context:
This question involves a geometric type number series in which each term is obtained from the previous one by multiplying by a constant factor. The presence of alternating positive and negative values, together with decreasing magnitudes, strongly suggests multiplication by a negative fraction. Our task is to find the missing term in the sequence 400, -200, 100, ?, 25, -12.5.

Given Data / Assumptions:
- Series terms: 400, -200, 100, ?, 25, -12.5.- Exactly one central term is missing.- Signs alternate positive and negative as we move along the series.- Magnitudes appear to be halving with each step.

Concept / Approach:
For geometric series, we examine the ratio of each term to the previous one. If the ratio remains constant, we can use it to generate missing terms. The alternating signs indicate that the common ratio is negative. The consistent halving of magnitudes suggests that the numerical value of the ratio is 1/2.

Step-by-Step Solution:
- From 400 to -200: ratio = -200 / 400 = -1/2.- From -200 to 100: ratio = 100 / (-200) = -1/2.- Thus the common ratio for the series is -1/2.- To find the missing term after 100, multiply 100 by -1/2.- 100 * (-1/2) = -50.- Check the remaining terms: -50 * (-1/2) = 25 and 25 * (-1/2) = -12.5, which matches the given series.

Verification / Alternative check:
- Rewriting the complete series using the ratio -1/2: 400, -200, 100, -50, 25, -12.5.- Each step multiplies the previous term by -1/2, so the pattern holds throughout.- There is no inconsistency once -50 is inserted, confirming it as the correct term.

Why Other Options Are Wrong:
- 50, 75, -75 and 25 do not satisfy the constant ratio -1/2 when placed between 100 and 25.- For example, if the missing term were 50, then 50 / 100 = 1/2 (not -1/2), and 25 / 50 = 1/2, breaking both the sign alternation and the geometric pattern.

Common Pitfalls:
- Some candidates check only differences instead of ratios, which is misleading in geometric series.- Ignoring the alternating signs can lead to a mistaken ratio of 1/2 instead of -1/2.- Confusion between division and multiplication by a fraction may cause computational errors.

Final Answer:
Each term is obtained by multiplying the previous one by -1/2, so the missing term is -50.