An arithmetic sequence is a sequence that has a constant difference between consecutive terms. That is, the difference between the first and the second term is the same as the difference between the second and the third term, and so on. This difference is called the common difference and is usually denoted with d. Consider the following example.
For this sequence, the common difference is d=2. Furthermore, consider the following example and note that the common difference can also be negative if an arithmetic sequence is decreasing.
In a theater, there are 10 rows of seats. The first row has 11 seats, and for each subsequent row, the number of seats increases by 3. Write an arithmetic sequence to represent the number of seats in each row, then graph the sequence.
Lastly, to graph the sequence, we can plot the points from the table. Notice that the points aren't connected. This is because the row number and the number of seats in each row both have to be whole numbers.
Because their terms change by a constant amount, arithmetic sequences show a linear relationship. Here, the common difference d=1.5 can be considered the slope of the line. In fact, when plotting an arithmetic sequence in a coordinate plane, it resembles the graph of a linear function.
To find the twelfth term of the sequence, we can now substitute n=12 into the rule.
The twelfth term is 37.
To begin, we must determine the common difference, d. Since we do not know the values of two consecutive terms, we cannot directly find d. However, the terms a3 and a6 are 3 positions apart, so they must differ by 3d.
Pelle is an avid collector of pellets. During his spare time, he likes to arrange his pellets in different patterns. Today, he's chosen to place them in the shape of a triangle. The top row consists of one pellet, the second of three pellets, the third of five pellets, and so on. Write a rule where is the amount of pellets in row n. Then, use the rule to find which row has 53 pellets.
To begin, we can make sense of the given information. For every row, the amount of pellets increase by 2. Thus, we know that d=2. It is also given that the first row consists of 1 pellet, a1=1. Using this information, we can find the rule.
Row 27 has 53 pellets.