Arithmetic sequence

Definition

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, and it plays a crucial role in determining the behavior and properties of the sequence. Each term can be expressed in terms of the first term and the number of terms, allowing for easy calculation of any term in the sequence.

5 Must Know Facts For Your Next Test

1. The general form for finding the nth term of an arithmetic sequence is given by $$a_n = a_1 + (n-1)d$$, where $$a_n$$ is the nth term, $$a_1$$ is the first term, and $$d$$ is the common difference.
2. If the common difference is positive, the arithmetic sequence will be increasing; if it is negative, the sequence will be decreasing.
3. The sum of the first n terms of an arithmetic sequence can be calculated using the formula $$S_n = \frac{n}{2} \times (a_1 + a_n)$$ or $$S_n = \frac{n}{2} \times (2a_1 + (n-1)d)$$.
4. An arithmetic sequence can be represented graphically as a straight line when plotted on a coordinate plane with terms on one axis and their corresponding positions on another.
5. Arithmetic sequences are commonly used in various real-life situations, such as calculating payments over time or determining sequential patterns in data.

Review Questions

• How can you determine whether a given sequence is an arithmetic sequence?
  • To determine if a sequence is arithmetic, check if the difference between consecutive terms remains constant throughout the sequence. This means you would subtract each term from the next to see if they all yield the same result. If they do, then it qualifies as an arithmetic sequence and you can identify the common difference easily.
• Calculate the 10th term of an arithmetic sequence where the first term is 5 and the common difference is 3.
  • Using the formula for the nth term $$a_n = a_1 + (n-1)d$$, plug in the values: $$a_{10} = 5 + (10-1) imes 3$$. This simplifies to $$a_{10} = 5 + 27 = 32$$. Thus, the 10th term of this arithmetic sequence is 32.
• Analyze how changing the common difference in an arithmetic sequence affects its graphical representation.
  • Changing the common difference in an arithmetic sequence alters the slope of its graphical representation. A larger positive common difference results in a steeper upward slope, while a smaller or negative common difference will produce a flatter or downward slope. This visual representation reflects how quickly or slowly the terms increase or decrease, highlighting how critical the common difference is to understanding both the behavior and appearance of arithmetic sequences.

Related terms

common difference: The fixed amount that each term of an arithmetic sequence increases or decreases by when moving from one term to the next.

nth term: The general formula that represents the value of the nth term in a sequence, often expressed as a function of n.

geometric sequence: A sequence of numbers in which each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio.