This question tests your understanding of number series and patterns. The given analogy is 13 : 20 :: 17 : ? This means we need to find the relationship between 13 and 20, and then apply that same relationship to 17 to find the missing number.

Let's analyze the relationship between 13 and 20:

• Difference: 20 - 13 = 7
• Addition: 13 + 7 = 20

Now, let's apply the same relationship (adding 7) to 17:

• 17 + 7 = 24

Since 24 is not one of the options, let's explore another possible relationship. Let's consider the square numbers:

• The number 13 is close to \(4^2 = 16\)
• The number 20 is close to \(4.5^2 = 20.25\)
• Therefore, we could think of the pattern as approximate squares. Let's check if this pattern works.

Let's try a different approach. Let's look at the difference between consecutive numbers.

• The difference between 20 and 13 is 7.
• Let's assume that the difference between consecutive pairs is constant. Then the next number would be 17 + 7 = 24.
• However, since 24 is not an option, let's reconsider.

Let's consider the relationship as adding a number, then adding an increasing number. The options suggest an arithmetic progression is unlikely. It appears the relationship is not simple addition.

Let's examine the options. The most suitable pattern appears to be a combination of addition and another operation.

The correct answer, 28, isn't easily derivable from a simple arithmetic operation on 17 based on the 13:20 relationship. More complex mathematical patterns or a different type of logic would be needed to explain this connection definitively.

Without a more defined pattern, the provided answer of 28 remains unexplained. The question may contain an error or require a non-obvious or unconventional pattern recognition skill. Further investigation might be needed to find a logical solution connecting 13, 20, 17, and 28.