47.	2,5,20,32,47,...
a.	64
b.	62
c.	60
d.	65
e.	61


To find the next number in the sequence 2, 5, 20, 32, 47, let's analyze the pattern. We will look at the differences between the consecutive terms:

1. \( 5 - 2 = 3 \)
2. \( 20 - 5 = 15 \)
3. \( 32 - 20 = 12 \)
4. \( 47 - 32 = 15 \)

Now let's analyze the differences themselves:

- The first difference is \( 3 \).
- The second difference is \( 15 \).
- The third difference is \( 12 \).
- The fourth difference is \( 15 \).

We notice that the differences are not constant, but let's see if there is a secondary pattern. The differences between these differences are:

1. \( 15 - 3 = 12 \)
2. \( 12 - 15 = -3 \)
3. \( 15 - 12 = 3 \)

This suggests a secondary pattern: 12, -3, 3. Let's check if this continues:

4. \( 15 - 12 = 3 \)

So the pattern appears to repeat every 3 steps: 12, -3, 3.

Given the current sequence:
1. \( 2 \)
2. \( 5 \)
3. \( 20 \)
4. \( 32 \)
5. \( 47 \)

The differences so far: \( 3, 15, 12, 15 \).

The next difference would be calculated based on the repeating pattern (12, -3, 3):

The last difference was 15, so following the pattern (12, -3, 3):
Next difference = \( 15 + 12 = 27 \).

Adding this to the last term in the sequence:
\( 47 + 27 = 74 \).

But, this does not match any of the choices given. Let's recheck the differences more closely to confirm.

Based on the first differences: 
\( 3, 15, 12, 15 \).

The differences themselves (second level):
\( 12, -3, 3 \).

Let's apply this more directly:
47 (last term)
Expected difference by pattern:
Next = \( 47 + 15 = 62 \) (since the 4th and 2nd term differences are similar).

Thus, the next term in the sequence is:
\( 47 + 15 = 62 \).

So, the correct answer is:

**b. 62**

Would you like details or have any questions? Here are five questions you might ask next:

1. How do you determine the pattern in a sequence?
2. Can you explain how to solve geometric sequences?
3. What are arithmetic sequences and how do you identify them?
4. How do you handle sequences with alternating patterns?
5. Can you help me with finding the nth term of a sequence?

Tip: When analyzing sequences, look for both primary and secondary patterns to identify complex progressions.

Learn how to identify patterns in number sequences with a detailed analysis of the sequence 2, 5, 20, 32, 47. This problem involves recognizing primary and secondary patterns in sequences, suitable for high school students studying advanced arithmetic.

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