Analyzing the Number Series Pattern: 35, 51, 76, 112, ?

The question asks us to identify the number that should replace the question mark (?) in the given series: 35, 51, 76, 112, ?. To solve this type of problem, we need to find the underlying pattern or rule that connects the terms in the series.

Finding the Difference Between Consecutive Terms

A common approach for number series questions is to look at the differences between consecutive terms. Let's calculate the differences:

• Difference between the 2nd and 1st term: $51 - 35 = 16$
• Difference between the 3rd and 2nd term: $76 - 51 = 25$
• Difference between the 4th and 3rd term: $112 - 76 = 36$

We can summarize these differences in a table:

Identifying the Pattern in Differences

Now, let's look at the sequence of differences we found: 16, 25, 36. Can we see a pattern in these numbers?

Yes, these numbers are perfect squares:

• $16 = 4 \times 4 = 4^2$
• $25 = 5 \times 5 = 5^2$
• $36 = 6 \times 6 = 6^2$

The pattern of differences is a sequence of consecutive perfect squares starting from $4^2$.

Predicting the Next Term in the Series

Following the identified pattern, the next difference in the series should be the next perfect square after $6^2$, which is $7^2$.

• The next difference should be $7 \times 7 = 49$.

To find the next term in the original series (which replaces the question mark), we need to add this difference (49) to the last known term (112).

• Next term $= \text{Last term} + \text{Next difference}$
• Next term $= 112 + 49$
• Next term $= 161$

Conclusion: The Missing Number

Based on the pattern of differences being consecutive perfect squares ($4^2, 5^2, 6^2, \dots$), the next difference is $7^2 = 49$. Adding this difference to the last term (112) gives us the missing number.

So, the number that replaces the question mark (?) in the series 35, 51, 76, 112, ? is 161.

Revision Table: Solving Number Series

Additional Information on Number Series Questions

Number series questions test your logical reasoning ability. There are various types of patterns you might encounter, such as:

• Arithmetic Series: Constant difference between terms.
• Geometric Series: Constant ratio between terms.
• Difference Series: The differences between terms follow a pattern (as in this question).
• Double Difference Series: Differences between differences follow a pattern.
• Mixed Series: Combination of two or more series.
• Fibonacci or Similar Series: Each term is the sum or product of previous terms.

Practicing different types of series helps in quickly identifying the pattern during exams. Always start by calculating the differences; it's a very common pattern.