Number series (2^n − 1 pattern): 15, 31, 63, 127, 255, (…) Determine the next Mersenne-type number in the sequence.

Introduction / Context:The sequence resembles Mersenne numbers of the form 2^n − 1. Recognizing powers of two and subtracting one is a staple pattern in number series problems.

Given Data / Assumptions:

• Given terms: 15, 31, 63, 127, 255.
• Test whether each equals 2^n − 1 for increasing n.

Concept / Approach:Check: 2^4 − 1 = 15, 2^5 − 1 = 31, 2^6 − 1 = 63, 2^7 − 1 = 127, 2^8 − 1 = 255. The next would be 2^9 − 1.

Step-by-Step Solution:

Verification / Alternative check:Observe the near-doubling between terms, consistent with powers of two less one.

Why Other Options Are Wrong:

• 513, 517, 523: Not equal to 2^n − 1 for n = 9; 2^9 − 1 is 511.

Common Pitfalls:Assuming exact doubling; the sequence grows approximately doubling minus one, not pure geometric progression.

Final Answer:511