Finding the Pattern in Number Series

Let's analyse the given number series: 6, 24, 60, 120, 210, ? We need to find the number that replaces the question mark by identifying the underlying pattern in the series.

Analysing the Number Series Pattern

We can try to find a relationship between consecutive terms or relate each term to its position in the series. Let's examine the terms:

• The first term is 6.
• The second term is 24.
• The third term is 60.
• The fourth term is 120.
• The fifth term is 210.

Let's try to see if there is a pattern involving the product of consecutive numbers.

• $1 \times 2 \times 3 = 6$ (This matches the first term)
• $2 \times 3 \times 4 = 24$ (This matches the second term)
• $3 \times 4 \times 5 = 60$ (This matches the third term)
• $4 \times 5 \times 6 = 120$ (This matches the fourth term)
• $5 \times 6 \times 7 = 210$ (This matches the fifth term)

It appears that the pattern for the \(n\)-th term in the series (starting with \(n=1\) for the first term) is the product of three consecutive integers: \(n\), \(n+1\), and \(n+2\). So, the \(n\)-th term is given by \(n(n+1)(n+2)\).

Calculating the Next Term in the Series

Based on the identified pattern, the next term in the series is the 6th term. Using the formula \(n(n+1)(n+2)\) with \(n=6\):

6th term = \(6(6+1)(6+2)\)

6th term = \(6 \times 7 \times 8\)

6th term = \(42 \times 8\)

6th term = \(336\)

Alternatively, we can observe another pattern. Each term can be represented as \(n^3 - n\), where \(n\) starts from 2.

• For n=2: \(2^3 - 2 = 8 - 2 = 6\) (1st term)
• For n=3: \(3^3 - 3 = 27 - 3 = 24\) (2nd term)
• For n=4: \(4^3 - 4 = 64 - 4 = 60\) (3rd term)
• For n=5: \(5^3 - 5 = 125 - 5 = 120\) (4th term)
• For n=6: \(6^3 - 6 = 216 - 6 = 210\) (5th term)

Following this pattern, the next term corresponds to \(n=7\).

Next term = \(7^3 - 7\)

Next term = \(343 - 7\)

Next term = \(336\)

Both patterns lead to the same result.

Comparing with Options

The calculated next term in the series is 336. Let's check the given options:

The calculated value, 336, matches Option 1.

Conclusion

The number that replaces the question mark (?) in the series 6, 24, 60, 120, 210, ? is 336, as it follows the pattern \(n(n+1)(n+2)\) for \(n=6\) (or \(n^3-n\) for \(n=7\)).

Revision Table: Number Series Patterns

Understanding different types of number series patterns is crucial for solving such problems. Here is a brief overview:

Additional Information on Solving Number Series Questions

Solving number series problems requires keen observation and practice. Here are some tips:

• Look for differences: Calculate the difference between consecutive terms. If the differences are constant, it's an arithmetic series. If not, check the differences of the differences (second differences), and so on.
• Look for ratios: Calculate the ratio between consecutive terms. If the ratio is constant, it's a geometric series.
• Check for squares and cubes: See if the terms are close to perfect squares (\(1, 4, 9, 16, 25, 36, \dots\)) or cubes (\(1, 8, 27, 64, 125, 216, \dots\)). The pattern might be \(n^2 \pm a\) or \(n^3 \pm a\).
• Look for alternating patterns: Sometimes, the pattern alternates between two different operations or sequences.
• Consider products or sums: The terms might be the product or sum of consecutive numbers or terms from another simple series.
• Combinations: More complex series might involve a combination of patterns.

Practicing with various types of number series helps in quickly identifying the pattern during exams.