Understanding the Number Series Pattern

To find the missing number in a series, we often look for a pattern in how the numbers change from one term to the next. This can involve checking the difference, ratio, or some other mathematical relationship between consecutive terms.

Let's examine the given number series: 35, 46, 59, 74, (…), 110.

We start by calculating the differences between consecutive numbers:

• Difference between the second and first term: $$46 - 35 = 11$$
• Difference between the third and second term: $$59 - 46 = 13$$
• Difference between the fourth and third term: $$74 - 59 = 15$$

The differences we found are 11, 13, and 15.

Now, let's look for a pattern in these differences. The differences themselves form a sequence: 11, 13, 15. We can see that each subsequent difference is obtained by adding 2 to the previous difference:

• $$13 = 11 + 2$$
• $$15 = 13 + 2$$

This suggests that the differences between consecutive terms form an arithmetic progression with a common difference of 2.

Step-by-Step Calculation for the Missing Number

Following the pattern of the differences (11, 13, 15), the next difference in the sequence should be the current difference (15) plus 2.

• Next difference = $$15 + 2 = 17$$

The missing number is obtained by adding this next difference (17) to the last known term in the series (74).

• Missing number = $$74 + 17$$
• Missing number = $$91$$

To verify this, let's find the difference that would follow 17 in the difference sequence. It would be $$17 + 2 = 19$$. The term after the missing number is 110. Let's check if the missing number (91) plus this next difference (19) equals 110.

• $$91 + 19 = 110$$

Since $$91 + 19 = 110$$, the pattern holds true for the entire series. The missing number is indeed 91.

Revision Table: Number Series Analysis

Let's summarize the number series and the differences found:

The differences between consecutive terms are 11, 13, 15, 17, 19, which increase by 2 each time.

Additional Information: Types of Number Series

Number series questions can involve various types of patterns. Some common types include:

• Arithmetic Series: A series where the difference between consecutive terms is constant (e.g., 2, 4, 6, 8...).
• Geometric Series: A series where the ratio between consecutive terms is constant (e.g., 3, 6, 12, 24...).
• Difference Series: As seen in this question, the differences between consecutive terms form a pattern, often an arithmetic or geometric series themselves.
• Square/Cube Series: Terms are squares or cubes of natural numbers (e.g., $$1^2, 2^2, 3^2, \dots$$ or $$1^3, 2^3, 3^3, \dots$$).
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 0, 1, 1, 2, 3, 5, 8...).
• Mixed Series: Series combining two or more different patterns or types.

Identifying the type of pattern is key to solving number series problems.