Solving the Number Series Question

The question asks us to find the missing term in the given series: 33, 40, 48, 57, 67, ?

To solve a number series question, we need to identify the pattern or rule that governs the sequence of numbers. Let's look at the differences between consecutive terms.

Finding the Pattern in the Number Series

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

• Difference between the 2nd and 1st term: \(40 - 33 = 7\)
• Difference between the 3rd and 2nd term: \(48 - 40 = 8\)
• Difference between the 4th and 3rd term: \(57 - 48 = 9\)
• Difference between the 5th and 4th term: \(67 - 57 = 10\)

We can organize these differences in a table to see the pattern clearly.

Looking at the differences (7, 8, 9, 10), we can observe a clear pattern. The differences are increasing by 1 each time.

Predicting the Next Difference and Term

Following the pattern of differences (7, 8, 9, 10), the next difference in the series should be \(10 + 1 = 11\).

To find the missing term, we need to add this next difference to the last known term in the series (67).

Missing term = Last term + Next difference

Missing term = \(67 + 11\)

Missing term = \(78\)

Therefore, the next term that completes the series is 78.

Checking the Options

Let's compare our calculated term with the given options:

1. 80
2. 76
3. 78
4. 81

Our calculated missing term, 78, matches option 3.

Revision Table: Key Steps for Solving Number Series

Additional Information on Number Series Patterns

Number series questions often involve various types of patterns. Here are a few common ones:

• Arithmetic Series: A constant difference between consecutive terms (e.g., 2, 4, 6, 8...).
• Geometric Series: A constant ratio between consecutive terms (e.g., 2, 4, 8, 16...).
• Difference Series: The differences between terms follow their own pattern (like in this question, the differences increase by 1). The differences themselves might be an arithmetic series, geometric series, squares, cubes, etc.
• Alternating Series: Two different patterns alternating within the same series.
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 0, 1, 1, 2, 3, 5, 8...).
• Series based on squares or cubes: Terms might be squares, cubes, or related to them (e.g., \(1^2, 2^2, 3^2...\) or \(1^3, 2^3, 3^3...\) or \(n^2 \pm k\), \(n^3 \pm k\)).

Practicing different types of series helps in quickly identifying the underlying pattern in exams.