To determine the next term in the sequence: 7, 41, 177, 721, 2897, let's examine the differences between consecutive terms.

1. Difference between 41 and 7:
   $41 - 7 = 34$

2. Difference between 177 and 41:
   $177 - 41 = 136$

3. Difference between 721 and 177:
   $721 - 177 = 544$

4. Difference between 2897 and 721:
   $2897 - 721 = 2176$

The pattern of differences is: $34, 136, 544, 2176$

These differences are each multiplied by 4 as we progress: $136 = 34 \times 4,\quad 544 = 136 \times 4,\quad 2176 = 544 \times 4$

Continuing this pattern, the next difference should be: $2176 \times 4 = 8704$

Adding this to the last term in the sequence (2897): $2897 + 8704 = 11601$

Thus, the next term in the sequence is 11601.

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Related Questions:

1. What is the formula for generating the terms of this sequence?
2. How would you recognize sequences with similar patterns?
3. Can you generalize this type of sequence to higher-order differences?
4. What would the next two terms of this sequence be?
5. How are geometric progressions related to sequences with multiplicative differences?

Tip:

Identifying the pattern in the differences between terms is a common technique when analyzing sequences.