Powers of two minus one Find the next term: 15, 31, 63, 127, 255, ( ... ).

Introduction / Context:This sequence consists of numbers of the form 2^n − 1. Recognizing these Mersenne-type values enables immediate continuation by increasing n by one and computing the next term.

Given Data / Assumptions:

• Terms: 15, 31, 63, 127, 255.
• Forms: 2^4 − 1 = 15, 2^5 − 1 = 31, 2^6 − 1 = 63, 2^7 − 1 = 127, 2^8 − 1 = 255.

Concept / Approach:Continue the pattern with n = 9 to get 2^9 − 1. Compute 2^9 = 512, then subtract 1 to obtain the next value.

Step-by-Step Solution:Identify n values: 4, 5, 6, 7, 8.Next exponent n = 9.Compute 2^9 = 512; subtract 1 ⇒ 511.

Verification / Alternative check:The ratio between successive terms approaches 2 as n grows, and differences roughly double, consistent with powers of two minus one. The exact continuation 511 is standard.

Why Other Options Are Wrong:513, 517, and 523 are not equal to 2^9 − 1; 383 corresponds to 2^8 − 1 + 128, not part of this pattern.

Common Pitfalls:Using addition or multiplication tables instead of identifying the exponential structure. Always test whether terms align with a simple power-based form.

Final Answer:511