Solving Number Series Patterns

This question asks us to find the next term in the given number series: 21, 55, 19, 45, 17, 35, ?. To solve number series questions, we look for a pattern between consecutive terms or between alternating terms.

Identifying the Alternating Series Pattern

Looking closely at the series 21, 55, 19, 45, 17, 35, ?, we can see that the numbers don't follow a simple linear or multiplicative pattern from one term to the next. However, if we look at alternate terms, we can identify two separate patterns or sub-series running within the main series.

Let's separate the terms into two sub-series:

1. Terms at odd positions: 1st, 3rd, 5th, 7th, ...
2. Terms at even positions: 2nd, 4th, 6th, ...

Analyzing the First Sub-series (Odd Positions)

The terms at the odd positions are 21, 19, 17, ?. Let's look for a pattern in this sub-series:

• From the 1st term (21) to the 3rd term (19): The difference is \(21 - 19 = 2\). The term decreases by 2.
• From the 3rd term (19) to the 5th term (17): The difference is \(19 - 17 = 2\). The term decreases by 2.

The pattern in the first sub-series is that each term is obtained by subtracting 2 from the previous term in this sub-series. This is an arithmetic progression with a common difference of -2.

The next term required is at the 7th position, which belongs to this first sub-series. Following the pattern, the 7th term will be the 5th term minus 2.

Next term = 17 - 2 = 15

Analyzing the Second Sub-series (Even Positions)

The terms at the even positions are 55, 45, 35. Let's look for a pattern in this sub-series:

• From the 2nd term (55) to the 4th term (45): The difference is \(55 - 45 = 10\). The term decreases by 10.
• From the 4th term (45) to the 6th term (35): The difference is \(45 - 35 = 10\). The term decreases by 10.

The pattern in the second sub-series is that each term is obtained by subtracting 10 from the previous term in this sub-series. This is an arithmetic progression with a common difference of -10.

Although the question asks for the 7th term (an odd position), understanding the pattern in the second sub-series confirms the alternating nature of the main series.

Determining the Next Term

The next term in the series is the 7th term, which falls in the first sub-series (21, 19, 17, ?). Based on our analysis of the first sub-series, the pattern is a decrease of 2 for each subsequent term in that sub-series.

The last term in the first sub-series provided is 17. The next term will be:

\(17 - 2 = 15\)

Thus, the missing term in the series is 15.

Conclusion on the Series Pattern

The given series is an alternating series composed of two interleaved arithmetic progressions:

• Series 1: 21, 19, 17, 15, ... (common difference -2)
• Series 2: 55, 45, 35, 25, ... (common difference -10)

The main series combines these as: 21 (S1), 55 (S2), 19 (S1), 45 (S2), 17 (S1), 35 (S2), 15 (S1), ...

The next number to complete the series 21, 55, 19, 45, 17, 35, ? is 15.

Revision Table: Number Series Concepts

Additional Information: Strategies for Solving Number Series

When faced with a number series question, try these common strategies:

• Check differences: Find the difference between consecutive terms. Is it constant? Increasing? Decreasing? Following a pattern?
• Check ratios: Find the ratio between consecutive terms. Is it constant?
• Check alternate terms: See if there's a pattern in terms at odd positions and another pattern in terms at even positions (alternating series).
• Look for squares, cubes, prime numbers: Sometimes the terms are related to squares, cubes, or prime numbers.
• Combine operations: The pattern might involve a combination of operations (e.g., multiply by 2, then add 1).
• Look at differences of differences: If the first differences don't show a clear pattern, look at the differences between those differences.

Applying the 'check alternate terms' strategy was key to solving this specific number series problem.