In the series 1000, 1012, 988, 1024, 978, which number is the odd man out if the intended pattern is to add and subtract increasing multiples of 12?

Introduction / Context:
This series alternates between rising and falling values: 1000, 1012, 988, 1024, 978. The jumps between terms suggest a pattern involving fixed multiples of 12, with the magnitude of the multiple increasing step by step. Our task is to see which term does not fit this add and subtract structure.

Given Data / Assumptions:
The terms of the series are:

• 1000
• 1012
• 988
• 1024
• 978

We assume that the intended pattern is to alternately add and subtract multiples of 12, with the multiples being 12, 24, 36 and 48 in order.

Concept / Approach:
We will compute the differences between consecutive terms and check whether these differences match ±12, ±24, ±36 and ±48. If four steps follow this rule and one does not, then the term involved in the incorrect step is the odd one out.

Step-by-Step Solution:
Step 1: From 1000 to 1012 the change is 1012 − 1000 = +12, which is +1 * 12.Step 2: From 1012 to 988 the change is 988 − 1012 = −24, which is −2 * 12.Step 3: From 988 to 1024 the change is 1024 − 988 = +36, which is +3 * 12.Step 4: Up to this point, we have added 12, subtracted 24 and added 36, so the next logical step is to subtract 48 (that is −4 * 12).Step 5: Subtracting 48 from 1024 gives 1024 − 48 = 976.Step 6: However, the given last term is 978, not 976, so 978 does not fit the pattern of adding and subtracting exact multiples of 12.

Verification / Alternative check:
We can reconstruct the intended series if no error is present. Starting from 1000 and applying the steps +12, −24, +36 and −48 gives 1000, 1012, 988, 1024 and 976. This sequence obeys the rule of alternately adding and subtracting increasing multiples of 12. Comparing this to the given series shows that all terms match except the last value, which should be 976 rather than 978.

Why Other Options Are Wrong:
The intermediate terms 1012, 988 and 1024 all fit the exactly computed differences of +12, −24 and +36. The starting term 1000 is fixed. Only the final term, 978, fails to be exactly 48 less than 1024, so it is the only one that breaks the rule. Therefore 978 must be the odd one out.

Common Pitfalls:
Because 978 is close to 976, it is easy to overlook the error and think that rounding or an approximate pattern is intended. However, number series questions usually rely on exact arithmetic, especially when dealing with clean multiples such as 12, 24, 36 and 48. Always check differences precisely rather than estimating.

Final Answer:
The number that does not follow the alternating plus and minus multiples of 12 pattern is 978.