Final answer:
For the expression A = 3n + 7, with n taking on the values 1, 2, and 3, the lower and greatest lower bound of A is 10. As A can technically grow indefinitely with larger n values (despite not being directly stated in the question), the upper bound and the least upper bound could be considered infinity.
Step-by-step explanation:
The student is asked to find the bounds for the expression A = 3n + 7 for n = 1, 2, 3. First, we need to calculate the value of A for each value of n, then determine the bounds.
- For n = 1: A = 3(1) + 7 = 10
- For n = 2: A = 3(2) + 7 = 13
- For n = 3: A = 3(3) + 7 = 16
The lower bound of A is the smallest value A can take, which is when n = 1, so A = 10. The greatest lower bound (also known as the infimum) of A is the same as the lower bound in this case, which is again 10. The upper bound of A is a value that A does not surpass; since n only takes on the values 1, 2, and 3, there is no stated upper limit, so we can say it is infinity. The least upper bound (also known as the supremum) of A is the smallest value that is greater than or equal to all possible values of A; since 16 is the highest value A achieves in the given range for n, the least upper bound would be 16.
However, the options provided do not offer 16 as an upper bound, which likely indicates that we're meant to consider open-ended boundaries rather than exact bounds. Therefore, since A can be indefinitely large if n were to continue beyond 3 (although that's not stipulated in the question), we could interpret the upper bounds, both least and greatest, as infinity