In the number series 6, 78, 858, 7722, 54055, 270270, which term is incorrect?

Introduction / Context:
This is an odd term in a number series question. The task is to detect the underlying pattern that generates the series and then check which term violates that pattern. These questions are common in competitive exams because they test pattern recognition, logic, and comfort with multiplication.

Given Data / Assumptions:

• Series given: 6, 78, 858, 7722, 54055, 270270.
• Exactly one term is wrong, all others follow a consistent rule.
• All numbers are positive integers.

Concept / Approach:
For rapidly increasing series, a sensible approach is to check multiplicative patterns. We can try to express each term as the previous term multiplied by some factor, possibly a sequence of related factors such as descending odd numbers or a regular progression.

Step-by-Step Solution:
Step 1: Calculate ratios using multiplication: 6 to 78 can be checked as 6 * 13 = 78. Step 2: From 78 to 858: 78 * 11 = 858. Step 3: From 858 to 7722: 858 * 9 = 7722. Step 4: From 7722 to 54055 as written: 7722 * 7 = 54054, but the term in the series is 54055, which is one more than 54054. Step 5: From 54054 to 270270: 54054 * 5 = 270270, which matches the last term exactly.

Verification / Alternative check:
We see that the pattern is: multiply successively by 13, 11, 9, 7, 5. If we generate the correct series using this rule we obtain 6, 78, 858, 7722, 54054, 270270. Therefore every step except the fourth is consistent with this rule, and the correct fourth product is 54054, not 54055. Since only a single term is allowed to be incorrect, 54055 is the odd term.

Why Other Options Are Wrong:
270270 is correct as 54054 * 5. The term 7722 is correct as 858 * 9. The term 858 is also correct as 78 * 11. None of these deviates from the pattern of multiplying by descending odd numbers. Only 54055 fails to equal 7722 multiplied by 7.

Common Pitfalls:
Learners sometimes divide adjacent terms and round ratios, which can obscure an exact integer pattern. Another pitfall is looking only at digit patterns instead of full multiplication. For fast growing series, always first check whether each term is an exact product of the previous term and a simple integer factor like 2, 3, 5, or a sequence such as 13, 11, 9, 7, 5.

Final Answer:
The term that does not follow the correct multiplication pattern is 54055.