91Ó°ÊÓ

An explicit formula allows you to determine any term in a sequence directly, without needing to know any previous terms. For arithmetic sequences, which have a constant difference between consecutive terms, the explicit formula is very straightforward.

In an arithmetic sequence, the explicit formula is given by:

• \(a_n = a_1 + (n - 1) \, \cdot \, d\)

Here, \(a_n\) represents the nth term in the sequence, \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.
This formula helps in calculating the value of any term in the sequence without listing all previous ones, which is efficient and faster.

For example, in the sequence from our exercise \(-5, -3.5, -2, -0.5, 1, \ldots\), we identified \(a_1 = -5\) and \(d = 1.5\). Thus, the explicit formula becomes:

• \(a_n = -5 + (n - 1) \, \cdot \, 1.5\)

While an explicit formula tells you how to compute a term directly, a recursive formula defines each term using its preceding term(s). This type of formula is useful in sequences, especially when the relationship between terms is consistent.

For arithmetic sequences, the recursive formula is:

• \(a_n = a_{n-1} + d\)

In this equation, \(a_n\) is the current term, \(a_{n-1}\) is the previous term, and \(d\) is the common difference.
This means to get the nth term, you add the common difference to the term before it.

In our example sequence \(-5, -3.5, -2, -0.5, 1, \ldots\), where \(d = 1.5\), the recursive form is:

• \(a_1 = -5\)
• \(a_n = a_{n-1} + 1.5\)

You begin with the first term \(a_1\) and apply the formula to find subsequent terms.

The common difference is a crucial element of arithmetic sequences, determining the consistent gap between consecutive terms. In every arithmetic sequence, this difference remains the same as you move from one term to the next.

When you examine a sequence, like \(-5, -3.5, -2, -0.5, 1, \ldots\), you calculate the common difference \(d\) by subtracting any term from the one that directly follows it.
For instance, in our sequence:

• \(-3.5 - (-5) = 1.5\)
• \(-2 - (-3.5) = 1.5\)

The difference \(1.5\) is consistent across consecutive terms, confirming it's an arithmetic sequence.

This consistency simplifies the process of understanding and working with the sequence, as you can use the common difference to construct both explicit and recursive formulas efficiently, allowing you to predict subsequent numbers in the sequence accurately.