91Ó°ÊÓ

An explicit formula in the context of arithmetic sequences provides a direct way to find any term in the sequence without needing to calculate all preceding terms. In our problem, the explicit formula given is \(a_{n} = -3 - 3n\). This formula defines each term \(a_n\) in the sequence based on its position \(n\).
The explicit formula here is linear, meaning it forms a straight line if plotted graphically with \(n\) on one axis and \(a_n\) on the other. This quality makes it particularly useful because any term in the sequence can be found quickly by simply substituting \(n\) into the formula.
For example, to find \(a_0\), simply replace \(n\) with 0 to get \(-3\), and similarly for \(a_1\) and \(a_2\) by substituting \(n\) with 1 and 2, respectively. This straightforward calculation shows the power and flexibility of the explicit formula.

Calculating terms in an arithmetic sequence using an explicit formula is like using a recipe: follow the steps, substitute the ingredients, and get the result.
Here's how you can do it:

• Identify the position \(n\) of the term you want to find, such as \(a_0\), \(a_1\), or \(a_2\).
• Substitute this \(n\) value into the explicit formula \(a_{n} = -3 - 3n\).
• Solve the resulting equation step by step.

For example, to find \(a_0\), substitute \(n = 0\) into the formula, which simplifies to \(a_0 = -3 - 3(0)\). This equals \(a_0 = -3\).
Repeating this process for other terms like \(a_1\) or \(a_2\), follows the same simple substitution and arithmetic steps, producing consistent results every time.

Arithmetic sequences have a distinctive pattern where the difference between consecutive terms is constant. This sequence's pattern is determined by the coefficient of \(n\) in the explicit formula \(a_{n} = -3 - 3n\).
The \(-3n\) part of the formula tells us that each term is 3 less than the previous one. This consistent decrement is the essence of the arithmetic sequence pattern.
Look at the terms we calculated:

• \(a_0 = -3\)
• \(a_1 = -6\)
• \(a_2 = -9\)

Notice how each term decreases by 3 as we move from one to the next. Understanding this pattern can help predict future terms and also verify calculated terms for accuracy. With an arithmetic sequence, once you know the starting point and the `common difference`, you can unlock the entire sequence using pattern recognition.