Analyzing the Number Series to Find the Incorrect Term

The given number series is 3, 7, 7, 11, 9, 13, 13, 17, 15, 23.

To find the incorrect number in the series, we need to identify the pattern that the numbers follow. Sometimes, a series might consist of two interleaved sub-series.

Let's examine the numbers at odd positions and even positions separately.

Series at Odd Positions

The numbers at the odd positions (1st, 3rd, 5th, 7th, 9th) are: 3, 7, 9, 13, 15.

Let's look at the difference between consecutive terms in this sub-series:

• $7 - 3 = 4$
• $9 - 7 = 2$
• $13 - 9 = 4$
• $15 - 13 = 2$

The pattern for the odd-positioned series is adding 4, then adding 2, repeating ($+4, +2, +4, +2, \dots$). The numbers in this sub-series match this pattern perfectly: $3 \xrightarrow{+4} 7 \xrightarrow{+2} 9 \xrightarrow{+4} 13 \xrightarrow{+2} 15$.

Series at Even Positions

The numbers at the even positions (2nd, 4th, 6th, 8th, 10th) are: 7, 11, 13, 17, 23.

Let's look at the difference between consecutive terms in this sub-series:

• $11 - 7 = 4$
• $13 - 11 = 2$
• $17 - 13 = 4$
• $23 - 17 = 6$

If this series followed a similar pattern to the odd series (or a consistent pattern like $+4, +2, +4, +2$), the differences should be 4, 2, 4, 2. However, the last difference is 6 instead of 2.

Following the observed $+4, +2, +4$ pattern from the start of this sub-series:

• Starting with 7
• $7 + 4 = 11$
• $11 + 2 = 13$
• $13 + 4 = 17$
• The next term should be $17 + 2 = 19$.

The last number in the given even-positioned series is 23, but based on the pattern $+4, +2, +4, +2$, it should be 19.

Identifying the Incorrect Number

The odd-positioned series (3, 7, 9, 13, 15) follows the pattern $+4, +2$ correctly.

The even-positioned series (7, 11, 13, 17, 23) follows the pattern $+4, +2, +4$ correctly up to 17, but the last term 23 does not fit the expected $+2$ step ($17+2=19$).

Therefore, the number 23 in the original series is the incorrect one.

Conclusion

The number 23 is incorrect as it disrupts the pattern in the series formed by the even-positioned numbers.

Revision Table: Number Series Pattern Analysis

Additional Information on Number Series Questions

Number series questions are common in aptitude tests and competitive exams. They test your ability to identify patterns in sequences of numbers.

Common types of patterns include:

• Arithmetic progression (constant difference)
• Geometric progression (constant ratio)
• Series with increasing or decreasing differences
• Alternating patterns (like $+,-,+,-\dots$)
• Interleaved series (two or more patterns combined)
• Patterns based on squares, cubes, prime numbers, etc.
• Fibonacci-like series (sum of previous terms)

When solving number series problems, it's helpful to:

• Look at the difference between consecutive terms.
• Look at the ratio between consecutive terms.
• Examine alternate terms for interleaved series.
• Consider squares, cubes, or other mathematical properties of the numbers.

Practice with various types of series helps in quickly recognizing the underlying pattern.