Solving Number Series Questions

Let's analyze the given number series: 5, 10, 17, 26, 37, 50, ? We need to find the pattern to determine the next number in this number series.

Analyzing the Pattern in the Series

We can look at the differences between consecutive terms in the series to identify a pattern.

• Difference between the 2nd term (10) and the 1st term (5): \(10 - 5 = 5\)
• Difference between the 3rd term (17) and the 2nd term (10): \(17 - 10 = 7\)
• Difference between the 4th term (26) and the 3rd term (17): \(26 - 17 = 9\)
• Difference between the 5th term (37) and the 4th term (26): \(37 - 26 = 11\)
• Difference between the 6th term (50) and the 5th term (37): \(50 - 37 = 13\)

Let's list the differences we found:

5, 7, 9, 11, 13

Identifying the Pattern in Differences

Now, let's look at the pattern in these differences. We can find the difference between these differences:

• Difference between the 2nd difference (7) and the 1st difference (5): \(7 - 5 = 2\)
• Difference between the 3rd difference (9) and the 2nd difference (7): \(9 - 7 = 2\)
• Difference between the 4th difference (11) and the 3rd difference (9): \(11 - 9 = 2\)
• Difference between the 5th difference (13) and the 4th difference (11): \(13 - 11 = 2\)

The differences between consecutive terms (the first differences) form a series where each term is 2 greater than the previous term. This is a consistent pattern. The sequence of differences is increasing by 2 each time.

Finding the Next Term in the Number Series

Following this pattern of differences, the next difference in the series 5, 7, 9, 11, 13 should be \(13 + 2 = 15\).

This next difference (15) is what we need to add to the last term of the original series (50) to get the next number.

Next term = Last term + Next difference

Next term = \(50 + 15\)

Next term = \(65\)

Therefore, the number that replaces the question mark (?) in the series is 65.

The next term is \(50 + 15 = 65\).

Revision Table: Number Series Patterns

Additional Information: How to Solve Number Series Questions

Solving number series problems often involves finding the underlying rule or pattern. Here are common strategies:

• Look for Differences: Calculate the differences between consecutive terms. If the differences are constant, it's an arithmetic series. If the differences form another simple pattern (like an arithmetic series themselves, as in this problem), it's a higher-order difference series.
• Look for Ratios: Divide each term by the previous term. If the ratio is constant, it's a geometric series.
• Look for Alternating Patterns: Sometimes patterns alternate (e.g., +2, -1, +2, -1).
• Look for Squares or Cubes: The terms might be related to square numbers ($n^2$), cube numbers ($n^3$), or these numbers plus/minus a constant.
• Combine Operations: The pattern might involve a combination of operations (e.g., multiply by 2 and add 1).
• Consider Position Number: The pattern might relate to the position of the term in the series (e.g., the nth term is $2n+1$).

For this specific number series problem, finding the differences was the key to uncovering the pattern and solving the question.