Analyzing the Number Analogy Pattern

The question asks us to find the missing term in the analogy 6 : 295 :: 7 : 1400 :: 8 : ?. This means the relationship between the first and second terms is the same as the relationship between the third and fourth terms, and this same relationship should be applied to the fifth term to find the sixth term.

Let's analyze the relationship between the given pairs of numbers:

1. First pair: 6 and 295
2. Second pair: 7 and 1400
3. Third pair: 8 and ?

Discovering the Relationship Pattern

We need to find a mathematical operation or pattern that connects 6 to 295 and 7 to 1400. Let's try exploring various mathematical operations involving the number itself (let's call it \(n\)).

Analyzing the First Pair (n=6)

For \(n=6\), the corresponding term is 295. Let's look at powers of 6:

• \(6^2 = 36\)
• \(6^3 = 216\)
• \(6^4 = 1296\)

The number 295 is relatively close to \(6^3 = 216\), but also quite a bit smaller than \(6^4 = 1296\). Let's consider the difference from \(6^4\):

\(1296 - 295 = 1001\)

This suggests a potential pattern involving \(n^4\) and a constant subtraction.

Analyzing the Second Pair (n=7)

For \(n=7\), the corresponding term is 1400. Let's look at powers of 7:

• \(7^2 = 49\)
• \(7^3 = 343\)
• \(7^4 = 2401\)

The number 1400 is relatively close to \(7^4 = 2401\). Let's check if the same difference we found for the first pair applies here:

\(2401 - 1001 = 1400\)

This matches the given second pair! The pattern seems to be \(n^4 - 1001\).

Applying the Pattern to the Third Pair (n=8)

Now that we have identified the pattern as \(n^4 - 1001\), we can apply it to the third term, which is \(n=8\), to find the missing term.

For \(n=8\), the corresponding term will be \(8^4 - 1001\).

First, calculate \(8^4\):

\(8^2 = 64\)

\(8^3 = 64 \times 8 = 512\)

\(8^4 = 512 \times 8 = 4096\)

Now, apply the subtraction:

\(8^4 - 1001 = 4096 - 1001\)

Let's perform the subtraction:

So, \(4096 - 1001 = 3095\).

The missing term in the analogy 8 : ? is 3095.

Verification

Let's quickly verify the pattern for all three terms:

• For 6: \(6^4 - 1001 = 1296 - 1001 = 295\). Correct.
• For 7: \(7^4 - 1001 = 2401 - 1001 = 1400\). Correct.
• For 8: \(8^4 - 1001 = 4096 - 1001 = 3095\). This is our calculated missing term.

The consistent pattern \(n^4 - 1001\) confirms that our calculation for the third pair is correct.

Final Answer

Based on the discovered pattern, the number related to 8 in the same way as 6 is related to 295 and 7 is related to 1400 is 3095.

Revision Table: Key Concepts

Additional Information: Number Reasoning Strategies

Number analogy and series questions often rely on recognizing patterns based on common mathematical operations. Here are some strategies students can use to tackle such problems:

• Check Basic Operations: Look for simple addition, subtraction, multiplication, or division between the terms.
• Consider Powers: Examine squares (\(n^2\)), cubes (\(n^3\)), or even fourth powers (\(n^4\)) of the numbers or numbers related to them (like \(n+1\) or \(n-1\)).
• Look for Differences/Ratios: Calculate the difference or ratio between consecutive terms or pairs to see if there is a constant value or a pattern in the differences/ratios.
• Combine Operations: Sometimes the pattern involves a combination of operations, such as \(An^p + B\), \(n^p \pm k\), \(n \times (n+k)\), etc.
• Prime Numbers, Squares, Cubes: Be familiar with prime numbers, perfect squares, and perfect cubes, as these often form the basis of patterns.
• Position-Based Patterns: In a series, the pattern might relate a term to its position in the sequence (e.g., \(n\) being the position number). In an analogy, \(n\) is typically the first number in the pair.
• Test Hypotheses: Once a potential pattern is identified, test it with all the given pairs to ensure consistency.

Practice with various types of number puzzles helps in quickly recognizing common patterns.