Find the 9th term of the AP, $18, 12, 6, 0, -62$
Answer (1)
Determine the common difference: d = 12 − 18 = − 6 .
Apply the formula for the nth term of an AP: a n = a 1 + ( n − 1 ) d .
Substitute the values: a 9 = 18 + ( 9 − 1 ) ( − 6 ) .
Calculate the 9th term: a 9 = − 30 .
Explanation
Understanding the Problem We are given an arithmetic progression (AP) and asked to find the 9th term. An arithmetic progression is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference. In this problem, we have the AP: 18 , 12 , 6 , 0 , − 6 , ...
Finding the Common Difference To find the 9th term, we first need to determine the common difference ( d ) of the AP. We can find d by subtracting any term from its subsequent term. Let's subtract the first term from the second term: d = 12 − 18 = − 6 .
Using the Formula for the nth Term Now that we have the common difference d = − 6 , we can use the formula for the n th term of an AP: a n = a 1 + ( n − 1 ) d , where a n is the n th term, a 1 is the first term, n is the term number, and d is the common difference.
Substituting the Values We are looking for the 9th term, so n = 9 . The first term is a 1 = 18 , and the common difference is d = − 6 . Substituting these values into the formula, we get: a 9 = 18 + ( 9 − 1 ) ( − 6 ) .
Calculating the 9th Term Now, we simplify the expression: a 9 = 18 + ( 8 ) ( − 6 ) = 18 − 48 = − 30 . Therefore, the 9th term of the AP is -30.
Final Answer Thus, the 9th term of the arithmetic progression is − 30 .
Examples
Arithmetic progressions are useful in various real-life scenarios, such as calculating simple interest, predicting salary increases, or determining the number of seats in a stadium row. For example, if you deposit $100 into a savings account that earns $5 simple interest each year, the amounts in your account each year form an arithmetic progression: $100, $105, $110, $115, and so on. Understanding arithmetic progressions helps you predict how your savings will grow over time.
