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An explicit formula is a mathematical expression that allows us to find any term in a sequence directly. It is a type of formula that clearly defines each term based on its position within the sequence, using a variable often denoted as \( n \).

In the given exercise, the explicit formula is \( a_n = 10^n + 3 \). This formula shows that for any term \( n \), you calculate the term by taking the base number 10, raising it to the power of \( n \), and adding 3. This type of formula is particularly useful because:

• It allows you to calculate any term without needing the previous terms.
• It provides a straightforward method to understand how the sequence progresses.

Using the explicit formula, you can quickly find terms no matter how large \( n \) is. For example, to find the 50th term, simply plug 50 into the formula without calculating terms one by one.

The term number in a sequence represents the position of a term. In mathematical sequences, this is often denoted by \( n \). Understanding the term number is crucial because the explicit formula relies on substituting \( n \) to find the corresponding term.

For instance, if you want to find the third term using the explicit formula \( a_n = 10^n + 3 \), you substitute \( n = 3 \). Knowing the term number helps determine:

• The calculation you need to perform.
• The order of the sequence, as each \( n \) provides a specific position.
• The relationship of the sequence terms with their positions.

By understanding the term's position, it becomes simpler to apply the explicit formula to find desired terms. So, always remember that \( n \) indicates the sequential location of each term.

Calculating the terms of a sequence involves plugging specific numbers for \( n \) into the explicit formula. Let's examine this process using the formula from the exercise, \( a_n = 10^n + 3 \):

• **First Term**: When \( n = 1 \), substitute into the formula to get \( a_1 = 10^1 + 3 = 13 \).
• **Second Term**: For \( n = 2 \), compute \( a_2 = 10^2 + 3 = 103 \).
• **Third Term**: With \( n = 3 \), find \( a_3 = 10^3 + 3 = 1003 \).
• **Fourth Term**: Finally, \( n = 4 \) yields \( a_4 = 10^4 + 3 = 10003 \).

Each calculation shows how substituting different values for \( n \) gives us different terms. This step-by-step method helps in understanding not only how to calculate individual terms but also in visualizing how the sequence grows. The calculations illustrate that as \( n \) increases, the terms become progressively larger, following the pattern dictated by the exponential growth inherent in the formula.