Solving the Number Series Problem

This question asks us to find the missing number in the given number series: 784, 568, 443, ?, 352. To solve this type of problem, we need to identify the pattern or rule that governs the sequence of numbers.

Analyzing the Given Number Series

Let's examine the relationship between consecutive terms in the series.

1. The first term is 784.
2. The second term is 568.
3. The third term is 443.
4. The fourth term is the missing number (?).
5. The fifth term is 352.

Finding the Pattern in the Series

Let's calculate the difference between consecutive terms:

• Difference between 784 and 568: $784 - 568 = 216$
• Difference between 568 and 443: $568 - 443 = 125$

The differences we found are 216 and 125. Let's think about these numbers. Do they relate to any common mathematical sequences like squares or cubes?

• We know that $6 \times 6 \times 6 = 6^3 = 216$.
• We know that $5 \times 5 \times 5 = 5^3 = 125$.

It appears that the differences between consecutive terms are cubes of decreasing integers. The first difference is $6^3$, and the second difference is $5^3$. Following this pattern, the next difference should be $4^3$, and the difference after that should be $3^3$.

Calculating the Missing Number

Based on the identified pattern, the difference between the third term (443) and the missing fourth term (?) should be $4^3$.

The pattern of differences is:

• Term 1 to Term 2: Subtract $6^3$ (216)
• Term 2 to Term 3: Subtract $5^3$ (125)
• Term 3 to Term 4: Subtract $4^3$ (64)
• Term 4 to Term 5: Subtract $3^3$ (27)

To find the missing number, we subtract $4^3$ from the third term:

Missing Number = Third Term - $4^3$

Missing Number = $443 - 64$

Missing Number = $379$

Verifying the Pattern

Let's check if the pattern holds for the next step using the calculated missing number (379). The difference between the missing term (379) and the fifth term (352) should be $3^3$.

Difference = Missing Number - Fifth Term

Difference = $379 - 352$

Difference = $27$

We know that $3 \times 3 \times 3 = 3^3 = 27$. Since the difference is 27, the pattern is confirmed.

The series with the missing number filled in is: 784, 568, 443, 379, 352.

The pattern is subtracting $6^3$, $5^3$, $4^3$, and $3^3$ respectively.

Summary of the Pattern and Solution

The pattern in the series is that each subsequent term is obtained by subtracting the cube of a decreasing integer, starting from 6, from the previous term.

$784 - 6^3 = 784 - 216 = 568$

$568 - 5^3 = 568 - 125 = 443$

$443 - 4^3 = 443 - 64 = 379$

$379 - 3^3 = 379 - 27 = 352$

Thus, the missing number is 379.

Revision Table: Number Series Patterns

Number series questions often follow various patterns. Reviewing common patterns can help you solve problems faster.

• Arithmetic Progression: Constant difference between consecutive terms.
• Geometric Progression: Constant ratio between consecutive terms.
• Difference Series: Differences between terms follow a pattern (e.g., arithmetic, geometric, squares, cubes, prime numbers).
• Alternating Series: Two different patterns alternate between terms.
• Squares/Cubes Series: Terms are squares or cubes of numbers, or related to them.
• Fibonacci Series: Each term is the sum of the two preceding terms (or a variation).

Additional Information: Tips for Solving Quantitative Aptitude Series

Here are some tips for tackling number series problems in quantitative aptitude tests:

• Look for differences between consecutive terms. See if these differences follow a pattern (arithmetic, geometric, squares, cubes, etc.).
• Look for ratios between consecutive terms.
• Check if the terms are related to squares or cubes of natural numbers.
• Consider alternating patterns.
• If the numbers are large, think about operations like division or subtraction of larger values. If the numbers are small and growing rapidly, think about multiplication or addition of larger values.
• Practice identifying different types of patterns. The more patterns you recognize, the easier it will be to spot them during an exam.