Final answer:
To find the common difference and the first term in the arithmetic sequence, use the nth term formula for the 100th and 200th terms, forming a system of equations to solve for the common difference, then back substitute to determine the first term.
Step-by-step explanation:
Given an arithmetic sequence where the 100th term is 12 and the 200th term is 89, we are asked to find the common difference in the sequence as well as the first term of the sequence. In an arithmetic sequence, the n-th term can be calculated using the formula T_n = a + (n-1)d, where T_n is the n-th term, a is the first term of the sequence, and d is the common difference.
To find the common difference, we can utilize the information that the 100th term (T_{100}) is 12 and the 200th term (T_{200}) is 89. By plugging these values into the n-th term formula for T_{100} and T_{200} and forming a system of equations, we can solve for the common difference d:
- T_{100} = a + (100-1)d = 12
- T_{200} = a + (200-1)d = 89
Subtracting equation (1) from equation (2), we eliminate a and can solve for d. The result gives us the common difference. Once the common difference is known, it can be substituted back into any of the two equations to solve for the first term a. The difference in the sequence refers to the common difference d, which is a crucial aspect in understanding the growth or decline of the values in the sequence.
Arithmetic sequences are a foundational concept in algebra and are essential for understanding how patterns and sequences work in mathematics. This step-by-step process of identifying terms of an arithmetic sequence is an essential skill for problem-solving in various mathematical applications.