Arithmetic Sequences

Definition

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference, and it allows the sequence to be generated by adding or subtracting the same value to each term.

5 Must Know Facts For Your Next Test

1. The common difference in an arithmetic sequence is denoted by the variable 'd'.
2. The explicit formula for an arithmetic sequence is $a_n = a_1 + (n-1)d$, where $a_n$ is the $n^{th}$ term, $a_1$ is the first term, and $d$ is the common difference.
3. The recursive formula for an arithmetic sequence is $a_n = a_{n-1} + d$, where $a_n$ is the $n^{th}$ term and $a_{n-1}$ is the '(n-1)^{th}' term.
4. Arithmetic sequences can be used to model real-world situations, such as the number of steps climbed in a staircase or the weekly salary of an employee with a constant raise.
5. The sum of the first $n$ terms of an arithmetic sequence can be calculated using the formula $S_n = \frac{n}{2}[2a_1 + (n-1)d]$.

Review Questions

• Explain how the common difference affects the terms in an arithmetic sequence.
  • The common difference in an arithmetic sequence is the constant value that is added or subtracted to each term to generate the next term. If the common difference is positive, the sequence will be increasing, and if the common difference is negative, the sequence will be decreasing. The magnitude of the common difference determines the rate of change between consecutive terms. A larger common difference will result in a faster rate of change, while a smaller common difference will result in a slower rate of change.
• Describe the relationship between the explicit formula and the recursive formula for an arithmetic sequence.
  • The explicit formula for an arithmetic sequence, $a_n = a_1 + (n-1)d$, allows you to calculate any term in the sequence directly using its position in the sequence, the first term, and the common difference. The recursive formula, $a_n = a_{n-1} + d$, on the other hand, allows you to calculate any term in the sequence using the previous term and the common difference. The recursive formula is useful for generating the sequence step-by-step, while the explicit formula is more efficient for finding a specific term.
• Analyze how the sum of the first $n$ terms of an arithmetic sequence can be used to model real-world situations.
  • The formula for the sum of the first $n$ terms of an arithmetic sequence, $S_n = \frac{n}{2}[2a_1 + (n-1)d]$, can be used to model various real-world situations. For example, it can be used to calculate the total distance traveled in a linear motion problem, the total number of steps climbed in a staircase problem, or the total salary earned by an employee with a constant raise over a certain number of pay periods. By understanding the relationship between the sum formula and the characteristics of the arithmetic sequence, you can use this knowledge to solve a wide range of practical problems.

Related terms

Common Difference: The constant difference between any two consecutive terms in an arithmetic sequence.

Explicit Formula: A formula that allows you to calculate any term in an arithmetic sequence using its position in the sequence.

Recursive Formula: A formula that allows you to calculate any term in an arithmetic sequence using the previous term and the common difference.