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In mathematics, a sequence is a list of numbers arranged in a specific order. Each number in the sequence is called a term. The sequence formula provides a way to find any term in the sequence using a mathematical expression. Here, the sequence formula given is \( a_n = 2 - \frac{1000}{n} \). This formula tells us that to find the \( n \)-th term of the sequence, subtract \( \frac{1000}{n} \) from 2.
The term \( n \) in the formula acts as a position indicator and as we plug different values for \( n \), we obtain different terms. For example, if \( n = 1 \), the term is \( a_1 = 2 - \frac{1000}{1} = -998 \). Similarly, with \( n = 2 \), \( a_2 = 2 - \frac{1000}{2} = -498 \).
Understanding how to use sequence formulas is fundamental in solving problems related to sequences and series.

Calculating a specific term in a sequence involves substituting the term number into the given sequence formula. Let's look at how we calculated the 100th term using the formula \( a_n = 2 - \frac{1000}{n} \).
First, we identify the given term number, which is 100. This means we need to find \( a_{100} \).

• Plug in the term number into the sequence formula: \( a_{100} = 2 - \frac{1000}{100} \).
• Next, simplify the fraction: \( \frac{1000}{100} = 10 \).
• Finally, perform the subtraction: \( 2 - 10 = -8 \).

Thus, \( a_{100} = -8 \).
Following these steps ensures that we accurately find the needed term in a sequence. The sequence formula simplifies this process by providing a direct method to calculate any term.

Simplification is a critical step in solving sequence problems. It involves reducing an expression or fraction to its simplest form to make further calculations easier.
In our sequence example, the simplification process was as follows:

• First, we had the expression \( a_{100} = 2 - \frac{1000}{100} \).
• We simplified the fraction \( \frac{1000}{100} \) to get 10.
• This simplified our expression to \( a_{100} = 2 - 10 \).
• Finally, subtracting 10 from 2 gave us \( -8 \).

Knowing how to simplify expressions is essential in sequence problems because it helps to find the correct term. It also makes complex expressions more manageable, eliminating errors in calculations.
Always remember, the goal of simplification is to break down complex sequences into more straightforward, easier-to-handle parts, ensuring accurate results.