Understanding the Number Series Question

The question asks us to find the missing number in the given series:

473, 464, 448, ?, 387

To solve a number series problem, we need to identify the pattern or rule that connects the terms in the sequence. This often involves looking at the differences between consecutive terms, ratios, or other mathematical operations.

Step-by-Step Analysis of the Number Series

Let's examine the differences between the consecutive terms provided in the series:

• Difference between the 1st and 2nd term: \(464 - 473 = -9\)
• Difference between the 2nd and 3rd term: \(448 - 464 = -16\)

The differences we have found are -9 and -16. Let's look closely at these numbers:

• \(9 = 3^2\)
• \(16 = 4^2\)

It appears that the differences between consecutive terms are negative perfect squares, starting from \(3^2\). Let's hypothesize that the pattern is subtracting consecutive perfect squares.

If this pattern holds, the next difference (between the 3rd term and the missing term) should be \(-5^2\).

• The 3rd term is 448.
• The next difference should be \(-5^2 = -25\).
• Let the missing term be \(x\). Then, \(x - 448 = -25\).

Calculating the Missing Number

Using the pattern identified, we can calculate the value of the missing term:

\(x - 448 = -25\)

Adding 448 to both sides of the equation:

\(x = 448 - 25\)

\(x = 423\)

So, the missing term is 423.

Verifying the Number Series Pattern

Let's check if the pattern continues with the calculated missing term and the last term of the series.

The series is now: 473, 464, 448, 423, 387.

• 473 to 464: \(473 - 9 = 473 - 3^2 = 464\) (Correct)
• 464 to 448: \(464 - 16 = 464 - 4^2 = 448\) (Correct)
• 448 to 423: \(448 - 25 = 448 - 5^2 = 423\) (Correct - This is the missing term)
• 423 to 387: The next difference should be \(-6^2 = -36\). Let's check: \(423 - 36 = 387\). (Correct)

The pattern of subtracting consecutive perfect squares (\(3^2, 4^2, 5^2, 6^2\)) holds true for the entire series. Therefore, the missing number is indeed 423.

Summary of the Pattern

The missing difference is \(-5^2 = -25\).

So, \(448 - 25 = 423\).

Conclusion on the Missing Term

Based on the consistent pattern of subtracting consecutive perfect squares starting from \(3^2\), the number that should come in place of the question mark is 423.

Revision Table: Number Series Patterns

Understanding common number series patterns is crucial for solving these types of questions. Here are a few examples:

• Arithmetic Series: Constant difference between terms (e.g., 2, 5, 8, 11... difference is 3).
• Geometric Series: Constant ratio between terms (e.g., 3, 6, 12, 24... ratio is 2).
• Difference Series: The differences between terms follow their own pattern (like this question, where differences were squares).
• Squares/Cubes Series: Terms are squares or cubes of numbers (e.g., 1, 4, 9, 16... are \(1^2, 2^2, 3^2, 4^2\)).
• Mixed Series: Combination of two or more patterns.
• Fibonacci Series: Each term is the sum of the two preceding ones (e.g., 1, 1, 2, 3, 5, 8...).

Additional Information: Approaching Number Series Problems

When faced with a number series question, consider these steps:

1. Calculate the differences between consecutive terms. See if there is a constant difference or a pattern in the differences.
2. Calculate the ratios between consecutive terms. See if there is a constant ratio.
3. Check for patterns involving squares, cubes, prime numbers, or other special number sequences.
4. Look for alternating patterns or patterns involving more than one previous term (like the Fibonacci series).
5. If the numbers are large or decrease significantly, consider subtraction or division patterns. If they increase, consider addition or multiplication patterns.

Practice with different types of series helps in quickly recognizing common patterns during exams.