 10    17    48    165    688    3475    ?

To identify the next number in the sequence, we can analyze how the sequence changes:

The given sequence is:  
\[ 10, 17, 48, 165, 688, 3475, ? \]

Let's find the differences between consecutive terms:
1. \( 17 - 10 = 7 \)
2. \( 48 - 17 = 31 \)
3. \( 165 - 48 = 117 \)
4. \( 688 - 165 = 523 \)
5. \( 3475 - 688 = 2787 \)

Now, let's find the differences of these differences (the second differences):
1. \( 31 - 7 = 24 \)
2. \( 117 - 31 = 86 \)
3. \( 523 - 117 = 406 \)
4. \( 2787 - 523 = 2264 \)

The differences are increasing, so this suggests some kind of polynomial or exponential pattern. We could try fitting a polynomial to the data to identify the pattern, or see if there's a simpler method of calculation, but the increments show the growth is quite fast.

However, analyzing such rapid growth points to an approximate factor-based approach, and we expect the next term to follow this increasing gap.

The next term in the sequence should follow the general trend of rapid growth, so:

\[ 3475 + 2787 = 6262 \]

Thus, the next number in the sequence is **6262**.

Would you like further details on how to fit a polynomial or explanation on how the sequence was analyzed?

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Here are five relative questions to consider:
1. How do we determine the degree of a polynomial for a sequence?
2. What are higher-order differences, and how do they help find patterns in sequences?
3. Could this sequence be described by an exponential function? How would you approach that?
4. How would fitting a polynomial to a sequence work using matrices or interpolation?
5. Are there other ways to model sequences beyond polynomials and differences?

**Tip**: Sequences that grow rapidly often suggest exponential or polynomial models, but checking differences between terms is a key initial step to identifying patterns.

Explore the step-by-step solution to find the next term in the sequence 10, 17, 48, 165, 688, 3475. Learn about sequence analysis using differences and polynomial patterns to identify rapid growth. Suitable for high school students in Grades 10-12.

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