- Identify the first term: \(a_1 = 64\).
- Determine the common difference: \(d = -5\).
- Write the recursive formula: \(a_1 = 64\), \(a_n = a_{n-1} - 5\).
- Write the explicit formula: \(a_n = 69 - 5n\).
- The final answer is \(a_1 = \boxed{64}, a_n = \boxed{a_{n-1}-5}\) and \(a_n = \boxed{69-5n}\).
Explanation
- Analyze the Arithmetic Sequence
The given sequence is 64, 59, 54, 49, 44, ... . An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is known as the common difference, denoted by \(d\). The first term of the sequence is denoted by \(a_1\). Here, the first term \(a_1\) is 64. To find the common difference \(d\), we subtract any term from the term that follows it: \(d = 59 - 64 = -5\). We can verify this with other terms: \(54 - 59 = -5\), \(49 - 54 = -5\), and so on. Thus, the common difference \(d\) is \(-5\).
- Determine the Recursive Formula
A recursive formula defines each term based on the preceding term. For an arithmetic sequence, the recursive formula is given by \(a_n = a_{n-1} + d\), where \(n > 1\). Substituting the identified first term and common difference, we have:
\[a_1 = 64\]
\[a_n = a_{n-1} - 5\]
This formula tells us that to get any term in the sequence, we take the previous term and subtract 5.
- Determine the Explicit Formula
The explicit formula allows us to find any term \(a_n\) directly by its position \(n\). The general form is \(a_n = a_1 + (n - 1)d\). Substituting our values \(a_1 = 64\) and \(d = -5\):
\[a_n = 64 + (n - 1)(-5)\]
Distributing the \(-5\):
\[a_n = 64 - 5n + 5\]
Combining the constant terms, we get:
\[a_n = 69 - 5n\]
This is the explicit formula for the sequence.
Examples
Arithmetic sequences are used in many real-life situations involving constant changes, such as saving money or planning schedules. For instance, if you start a savings account with \(\$64\) and add \(\$5\) less every month (meaning you deposit \(\$59\), then \(\$54\), etc.), the balance or the deposit amount follows an arithmetic pattern. By using the recursive or explicit formula, you can easily calculate how much you will have deposited in any given month without having to list every single month's deposit.