$a_n = a_1 + (n - 1)d$ is a formula used to describe the general term of an arithmetic sequence. It represents the $n^{th}$ term of the sequence, where $a_1$ is the first term, n is the term number, and d is the common difference between consecutive terms.
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The formula $a_n = a_1 + (n - 1)d$ allows you to find the $n^{th}$ term of an arithmetic sequence given the first term ($a_1$) and the common difference ($d$).
The term number ($n$) represents the position of the term within the sequence, starting from 1 for the first term.
The common difference ($d$) is the constant value added to each successive term to generate the next term in the sequence.
The formula can be used to find any term in the sequence, not just the $n^{th}$ term, by substituting the appropriate value for $n$.
Arithmetic sequences are widely used in various applications, such as finance, physics, and engineering, to model linear patterns of growth or decay.
Review Questions
Explain how the formula $a_n = a_1 + (n - 1)d$ is used to generate the terms of an arithmetic sequence.
The formula $a_n = a_1 + (n - 1)d$ is used to generate the terms of an arithmetic sequence by starting with the first term ($a_1$) and then adding the common difference ($d$) multiplied by the term number ($n - 1$) to get the $n^{th}$ term. This allows you to calculate any term in the sequence, as long as you know the first term and the common difference. For example, if the first term is 3 and the common difference is 2, then the 5th term would be $a_5 = 3 + (5 - 1)2 = 11$.
Describe how the parameters $a_1$ and $d$ in the formula $a_n = a_1 + (n - 1)d$ affect the overall behavior of an arithmetic sequence.
The parameters $a_1$ and $d$ in the formula $a_n = a_1 + (n - 1)d$ have a significant impact on the behavior of an arithmetic sequence. The first term ($a_1$) determines the starting point or initial value of the sequence, while the common difference ($d$) determines the rate of change between consecutive terms. If $d$ is positive, the sequence will exhibit linear growth, with each term being greater than the previous one. If $d$ is negative, the sequence will exhibit linear decay, with each term being less than the previous one. The combination of $a_1$ and $d$ determines the specific values and pattern of the sequence.
Analyze how the formula $a_n = a_1 + (n - 1)d$ can be used to solve problems involving arithmetic sequences, such as finding the $n^{th}$ term or the sum of the first $n$ terms.
The formula $a_n = a_1 + (n - 1)d$ is a powerful tool for solving a variety of problems involving arithmetic sequences. By rearranging the formula, you can use it to find the $n^{th}$ term of the sequence, given the first term ($a_1$) and the common difference ($d$). Additionally, the formula can be used to find the sum of the first $n$ terms of an arithmetic sequence by leveraging the relationship between the first and last terms, as well as the number of terms. This allows you to solve problems related to finance, physics, and other real-world applications that involve linear patterns of growth or decay.