Finding the Missing Term in the Number Series

The given series is: 1, 4, 3, 16, 6, 36, 10, 64, 15, 100, ?

To solve this number series problem, we need to identify the pattern governing the sequence of numbers. Observing the series, we can see that it alternates between relatively small numbers and larger numbers that appear to be perfect squares.

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

Pattern in Odd Positions

The terms at the odd positions (1st, 3rd, 5th, 7th, 9th) are: 1, 3, 6, 10, 15.

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

• Difference between 3rd and 1st term: \(3 - 1 = 2\)
• Difference between 5th and 3rd term: \(6 - 3 = 3\)
• Difference between 7th and 5th term: \(10 - 6 = 4\)
• Difference between 9th and 7th term: \(15 - 10 = 5\)

The differences between consecutive terms form a sequence: 2, 3, 4, 5. This sequence shows an increasing difference of 1 each time. Therefore, the next difference should be 6.

The missing term is at the 11th position, which is an odd position. To find the 11th term, we add the next difference (6) to the 9th term (15).

Next term in odd positions = 9th term + 6 = \(15 + 6 = 21\).

Pattern in Even Positions

The terms at the even positions (2nd, 4th, 6th, 8th, 10th) are: 4, 16, 36, 64, 100.

Let's look at these numbers:

• 2nd term: \(4 = 2^2\)
• 4th term: \(16 = 4^2\)
• 6th term: \(36 = 6^2\)
• 8th term: \(64 = 8^2\)
• 10th term: \(100 = 10^2\)

The terms at even positions are the squares of even numbers: 2, 4, 6, 8, 10. The base of the square increases by 2 each time.

The missing term is at the 11th position, which is an odd position, so this even position pattern does not directly give us the missing term. However, understanding this pattern confirms the alternating nature of the series.

Determining the Missing Term

The missing term is the 11th term in the series. Since 11 is an odd number, the missing term follows the pattern of the terms at odd positions.

Based on our analysis of the odd positions, the next term after 15 should be \(15 + 6 = 21\).

Thus, the missing term in the series is 21.

Let's check the options provided:

• 21
• 19
• 22
• 25

Our calculated missing term is 21, which matches one of the options.

Revision Table: Series Analysis Summary

Additional Information: Types of Number Series

Number series problems often appear in logical reasoning sections of competitive exams. Identifying the pattern is key. Common types of number series include:

• Arithmetic Series: Where the difference between consecutive terms is constant. Example: 2, 5, 8, 11, ... (difference is 3)
• Geometric Series: Where the ratio between consecutive terms is constant. Example: 3, 6, 12, 24, ... (ratio is 2)
• Square Series: Where terms are squares of numbers. Example: 1, 4, 9, 16, ... (\(1^2, 2^2, 3^2, 4^2, ...\))
• Cube Series: Where terms are cubes of numbers. Example: 1, 8, 27, 64, ... (\(1^3, 2^3, 3^3, 4^3, ...\))
• Difference Series: Where the differences between consecutive terms form a pattern (like in the odd positions of our problem).
• Alternating Series: Where two different patterns alternate, affecting terms at odd and even positions, or simply alternating in sequence. This is the type of series we solved here.
• Fibonacci Series: Where each term is the sum of the two preceding terms. Example: 0, 1, 1, 2, 3, 5, 8, ...

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