Number Series Logic: Finding the Wrong Term

The question asks us to identify the incorrect number in the given series: 3, 10, 27, 4, 16, 64, 5, 25, 125. We need to find the term that does not fit the pattern established by the other numbers.

Analyzing the Number Series Structure

Let's examine the series closely. It appears to be divided into distinct groups, or triplets, of numbers:

• First triplet: 3, 10, 27
• Second triplet: 4, 16, 64
• Third triplet: 5, 25, 125

Identifying the Pattern in Triplets

We will now analyze the relationship between the numbers within each triplet to find a consistent pattern.

Triplet 2 Analysis (4, 16, 64)

In the second triplet:

• The first number is 4.
• The second number is 16. This is the square of the first number, calculated as $4^2 = 16$.
• The third number is 64. This is the cube of the first number, calculated as $4^3 = 64$.

This triplet follows the pattern $n, n^2, n^3$, where $n=4$.

Triplet 3 Analysis (5, 25, 125)

In the third triplet:

• The first number is 5.
• The second number is 25. This is the square of the first number, calculated as $5^2 = 25$.
• The third number is 125. This is the cube of the first number, calculated as $5^3 = 125$.

This triplet also perfectly follows the pattern $n, n^2, n^3$, where $n=5$.

Applying the Pattern to Triplet 1

Now, we apply the same pattern, $n, n^2, n^3$, to the first triplet, where the starting number is 3.

For $n=3$, the expected numbers should be:

• First number: $n = 3$
• Second number: $n^2 = 3^2 = 9$
• Third number: $n^3 = 3^3 = 27$

Therefore, the first triplet should ideally be 3, 9, 27 to maintain the established pattern.

Determining the Wrong Term

Let's compare the actual given first triplet (3, 10, 27) with the expected triplet (3, 9, 27):

• The first term, 3, is correct.
• The second term given is 10. However, based on the pattern ($n^2$), it should be 9. This is where the series deviates.
• The third term, 27, is correct.

Since the number 10 does not fit the $n^2$ rule for the first triplet, it is the incorrect term.

Conclusion

The wrong term in the number series is 10. The series follows the pattern $n, n^2, n^3$ for consecutive integers $n=3, 4, 5$. The number 10 should be 9 to match this pattern.