Answer: (1) The scenario represents an arithmetic sequence because each time an omelet is made, the number of remaining eggs decreases by the same amount.
(2) Let's denote the number of remaining eggs after making the n-th omelet by a_n. We can see from the problem statement that the difference between any two consecutive terms in the sequence is constant. Let d be a common difference. Then, we have:
a_{n+1} = a_n - d
Using the information from the problem statement, we can find the value of d:
a_2 - a_1 = 174 - 177 = -3
a_3 - a_2 = 171 - 174 = -3
Since the difference is the same in both cases, we have d = -3. Therefore, the recursive formula is:
a_{n+1} = a_n - 3, with a_1 = 177.
(3) To find an explicit formula, we can use the recursive formula to derive a general expression for a_n. Starting with the recursive formula, we have:
a_2 = a_1 - 3
a_3 = a_2 - 3 = a_1 - 23
a_4 = a_3 - 3 = a_1 - 33
a_5 = a_4 - 3 = a_1 - 4*3
We can see that the general expression for a_n is:
a_n = a_1 - (n-1)*3
Substituting a_1 = 177, we get:
a_n = 180 - 3n
Therefore, the explicit formula for the number of remaining eggs after making the n-th omelet is a_n = 180 - 3n.
(4) To find the number of eggs Joe will have left after making 42 omelets, we can simply substitute n = 42 into the explicit formula:
a_{42} = 180 - 3*42 = 54
Therefore, Joe will have 54 eggs left after making 42 omelets.
Explanation: