The question asks us to find the number that replaces the question mark (?) in the given number series: 25, 36, 51, 70, 93, 120, ?.

To solve this number series problem, we need to identify the pattern or rule governing the sequence of numbers. Let's look at the differences between consecutive terms in the series.

Analyzing the Number Series Pattern

We calculate the difference between each term and the previous term:

• Difference between 36 and 25: $$36 - 25 = 11$$
• Difference between 51 and 36: $$51 - 36 = 15$$
• Difference between 70 and 51: $$70 - 51 = 19$$
• Difference between 93 and 70: $$93 - 70 = 23$$
• Difference between 120 and 93: $$120 - 93 = 27$$

The sequence of these first differences is 11, 15, 19, 23, 27.

Let's look at the differences between these first differences (the second differences):

• Difference between 15 and 11: $$15 - 11 = 4$$
• Difference between 19 and 15: $$19 - 15 = 4$$
• Difference between 23 and 19: $$23 - 19 = 4$$
• Difference between 27 and 23: $$27 - 23 = 4$$

The second differences are constant and equal to 4. This indicates that the pattern is based on adding increasing numbers, where the increase itself is constant. This is characteristic of a quadratic sequence.

Predicting the Next Number in the Sequence

Since the second difference is always 4, the next difference in the sequence of first differences (11, 15, 19, 23, 27) will be $$27 + 4 = 31$$.

To find the next number in the original series, we add this next difference (31) to the last term in the series (120).

Next term = Last term + Next difference

Next term = $$120 + 31 = 151$$

So, the number that replaces the question mark (?) is 151.

Summary of the Number Series and Differences

The pattern is consistent, with a constant second difference of 4. The next number in the series is 151.

Revision Table: Number Series Patterns

Additional Information on Finding Patterns in Sequences

Finding the pattern in a number series often involves looking at the differences between terms. If the first differences are constant, it's an arithmetic sequence. If the second differences are constant, it's a quadratic sequence. If the third differences are constant, it's a cubic sequence, and so on.

Sometimes, patterns can involve multiplication, division, squares, cubes, or combinations of operations. Examining the differences is a common first step for polynomial sequences (linear, quadratic, cubic, etc.).

In this specific number series (25, 36, 51, 70, 93, 120, ?), the identification of a constant second difference of 4 clearly points to it being a quadratic sequence, allowing us to predict the subsequent terms accurately.