Find the next term in the number series 125, 80, 45, 20, ....

Introduction / Context:
This question involves identifying a pattern in a number series. Many aptitude tests include such problems to check a candidate ability to detect simple arithmetic patterns and extend them to find missing terms.

Given Data / Assumptions:
The given series is 125, 80, 45, 20, ...We are required to find the next term in this series.

Concept / Approach:
A useful strategy is to examine the differences between consecutive terms and see whether these differences follow a pattern. If the first level differences themselves form a simple sequence, such as an arithmetic progression, we can use that to predict the next difference and therefore the next term.

Step-by-Step Solution:
Step 1: Compute the difference between the first and second terms: 80 − 125 = −45.Step 2: Compute the difference between the second and third terms: 45 − 80 = −35.Step 3: Compute the difference between the third and fourth terms: 20 − 45 = −25.Step 4: Observe the pattern in the differences: −45, −35, −25.Step 5: These differences themselves form an arithmetic sequence with common difference 10, increasing by 10 each time (−45, −35, −25, ...).Step 6: Therefore, the next difference should be −15, continuing the pattern.Step 7: Subtract this next difference from the last given term: 20 − 15 = 5.Step 8: Hence, the next term in the series is 5.

Verification / Alternative check:
We can reconstruct the series using the pattern of differences. Start with 125, subtract 45 to obtain 80, subtract 35 to obtain 45, subtract 25 to obtain 20 and then subtract 15 to obtain 5. The pattern is consistent, verifying that 5 is the correct next term.

Why Other Options Are Wrong:
The option 10 would arise if we mistakenly used a constant difference instead of a changing one, but that does not match the observed values. The option 15 or −5 does not align with the pattern of differences increasing by 10 in absolute value but becoming less negative each time. Only 5 maintains the observed trend of differences.

Common Pitfalls:
Some learners might try to find a multiplicative pattern rather than looking at differences, which is not appropriate here. Others may incorrectly compute one of the differences, which leads to an incorrect inference about the pattern. Careful calculation and checking of each difference is important.

Final Answer:
The next number in the series 125, 80, 45, 20, ... is 5.