Final answer:
The formulas provided are used to find the value of the nth term in different types of sequences: arithmetic, geometric, and a special case with given first and second terms. The formulas involve the first term, common difference/ratio, and the number of terms.
Step-by-step explanation:
The first formula, a_n = a_1 + (n-1)d, is used to find the value of the nth term in an arithmetic sequence. The formula represents the common difference (d) between consecutive terms. To find the first term (a_1) and the common difference (d), you need the given information or patterns in the sequence.
The second formula, a_n = a_1 * r^(n-1), is used to find the value of the nth term in a geometric sequence. The formula represents the common ratio (r) between consecutive terms. To find the first term (a_1) and the common ratio (r), you need the given information or patterns in the sequence.
The third formula, a_n = a_1 + (n-1)(a_2 - a_1), is used to find the value of the nth term in an arithmetic sequence given the first (a_1) and second (a_2) terms. It represents the difference between the second and first terms multiplied by the number of terms.
The fourth formula, a_n = a_1 * (n-1)!, is not commonly used to find the value of the nth term. It represents the factorial of (n-1) multiplied by the first term (a_1).