91Ó°ÊÓ

In an arithmetic sequence, each term is derived by adding a constant value, known as the 'common difference', to the previous term. This constant difference is key to understanding the pattern of the sequence. For the sequence given in the exercise, we calculated the common difference as follows:

• We know the first term: \(a_1 = -4\).
• We know the fifth term: \(a_5 = 16\).

To find the common difference \(d\), we used the formula: \[ d = \frac{a_5 - a_1}{5 - 1} \] Substituting the known terms: \[ d = \frac{16 - (-4)}{4} = \frac{20}{4} = 5 \] This means that each term in this sequence increases by 5 as we move from one term to the next. Understanding the common difference allows us to accurately chart the progression of terms within the sequence.

The nth term formula is a powerful tool in arithmetic sequences, allowing us to find any term without listing all previous ones. The nth term formula in an arithmetic sequence is given by:\[ a_n = a_1 + (n - 1) \cdot d \]This formula connects the position of a term in the sequence (\(n\)) with its value (\(a_n\)). In our exercise:

• \(a_1 = -4\), the first term,
• and \(d = 5\), the common difference.

Substituting these values into the nth term formula, we have:\[ a_n = -4 + (n - 1) \cdot 5 \]Simplifying further, it becomes:\[ a_n = -4 + 5n - 5 \]Finally, it simplifies to:\[ a_n = 5n - 9 \]With this formula, you can find any term in our sequence without direct calculation of all previous terms, simply by plugging in the value of \(n\).

Sequences are ordered lists of numbers following a specific pattern. They are fundamental in mathematics, often appearing in different branches like algebra, calculus, and beyond. Within sequences, each number is referred to as a "term." In arithmetic sequences, each term increases or decreases by a constant amount, known as the common difference.

• An arithmetic sequence is defined by its first term and its common difference.
• The sequence progresses by adding the common difference to obtain subsequent terms.

For example, if the first term is \(a_1 = -4\) and the common difference \(d = 5\), the sequence would be \(-4, 1, 6, 11, 16, \ldots\).
To determine a specific term or the general pattern, formulas like the nth term formula come into play. By understanding these core components—terms, common difference, and pattern—students can grasp how sequences are constructed and manipulated in math.