Final answer:
Based on the provided clues, it is suggested that the number of sticks in the nth pattern follows a quadratic relationship, expressed either as n2 or 2n2. This indicates a regular increase by squares of the pattern number, which is typical of a quadratic sequence rather than a linear or exponential one.
Step-by-step explanation:
To determine the rule for the number of sticks in the nth pattern, we look for a mathematical relationship that can be applied consistently across the patterns. In mathematics, a common way to identify such rules is to look for sequences where a fixed number is multiplied by itself, which can be expressed using exponents. If the number of sticks doubles each time, we may be looking at an exponential growth pattern, but since we are given that the same number of sticks is added each time, this suggests a linear relationship, which can often be defined by a simple arithmetic sequence.
Based on the reference to M = bn, where b is the base and n is the exponent, we might initially consider the relationship between pattern number and the number of sticks as an exponential one. However, the reference to n terms equaling n2 and the example of orbital angular momentum states suggests that for each increase in the pattern number n, the number of sticks corresponds to n2. It seems that the pattern uses squares of the pattern number to find the number of sticks.
In conclusion, the additional reference to 2n2 solidifies the idea that we are dealing with a quadratic pattern, and specifically that the rule for the number of sticks in the nth pattern is n2 or 2n2, depending on whether we see a doubling each time or just an addition of n2 sticks each time. If the latter, adding something like n sticks to the previous term, the relationship would be more likely to be a simple arithmetic sequence.