The start of an arithmetic sequence is
\[16, \quad 19, \quad 22, \quad 25, \quad \ldots\]
The rule for the sequence can be written in the form \[x_n=c n+d\], where \[c\] and \[d\] are numbers.
a) By first calculating the values of \[c\] and \[d\], work out the rule for the sequence.
b) What is the value of \[x_{11}\]?
Answer (1)
Find the common difference: c = 19 − 16 = 3 .
Substitute the first term into the formula to find d : 16 = 3 ( 1 ) + d , so d = 13 .
Write the rule for the sequence: x n = 3 n + 13 .
Calculate x 11 : x 11 = 3 ( 11 ) + 13 = 46 . The value of x 11 is 46 .
Explanation
Understanding the Problem We are given an arithmetic sequence 16 , 19 , 22 , 25 , … and we want to find a rule for the sequence in the form x n = c n + d . Then we want to find the value of x 11 .
Finding the Common Difference First, we need to find the common difference of the arithmetic sequence. This will be the value of c . The common difference is the difference between consecutive terms. So, c = 19 − 16 = 3 .
Finding the Value of d Now we know that c = 3 , so the rule is x n = 3 n + d . To find d , we can substitute the first term ( x 1 = 16 ) and n = 1 into the formula: 16 = 3 ( 1 ) + d .
Calculating d Solving for d , we get d = 16 − 3 = 13 .
Writing the Rule So the rule for the sequence is x n = 3 n + 13 .
Finding x_11 Now we want to find x 11 . We substitute n = 11 into the rule: x 11 = 3 ( 11 ) + 13 .
Calculating x_11 Calculating x 11 , we get x 11 = 33 + 13 = 46 .
Final Answer Therefore, the rule for the sequence is x n = 3 n + 13 , and the value of x 11 is 46.
Examples
Arithmetic sequences are useful in many real-world scenarios, such as predicting future values based on a pattern. For example, if you save $100 each month, the total amount saved over time forms an arithmetic sequence. Understanding the rule for the sequence allows you to easily calculate how much you'll have saved after a certain number of months. This concept is also applicable in calculating simple interest, where the interest earned each period is constant, forming an arithmetic sequence.
