Answer each question about the following arithmetic series:
[tex]12+18+24+30+\ldots+198[/tex]
What is the explicit rule for the arithmetic sequence on which the series is based?
[tex]a_n=12 +(n-1) 6[/tex]
How many terms are in the series?
[tex]n=[/tex]
Answer (1)
We have the arithmetic series 12 + 18 + 24 + 30 + … + 198 and the explicit rule a n = 12 + ( n − 1 ) 6 .
We set the last term equal to the explicit rule: 198 = 12 + ( n − 1 ) 6 .
We simplify and solve for n : 198 = 6 + 6 n ⇒ 192 = 6 n ⇒ n = 6 192 .
We find the number of terms: n = 32 , so the final answer is 32 .
Explanation
Understanding the Problem We are given an arithmetic series 12 + 18 + 24 + 30 + … + 198 . We need to find the number of terms in this series. We are also given the explicit rule for the arithmetic sequence on which the series is based: a n = 12 + ( n − 1 ) 6 .
Setting up the Equation The n -th term of the series is given by a n = 12 + ( n − 1 ) 6 . The last term of the series is 198, so we set a n = 198 and solve for n .
Simplifying the Equation We need to solve the equation 198 = 12 + ( n − 1 ) 6 for n . Let's simplify the equation:
198 = 12 + 6 n − 6 198 = 6 + 6 n
Isolating the Variable Subtract 6 from both sides of the equation:
198 − 6 = 6 n 192 = 6 n
Solving for n Divide both sides by 6 to solve for n :
n = 6 192 n = 32
Final Answer Therefore, there are 32 terms in the series.
Examples
Arithmetic series are useful in many real-life situations. For example, if you save a certain amount of money each month, the total amount you've saved over time forms an arithmetic series. Understanding how to calculate the number of terms helps you predict when you'll reach a savings goal. Another example is calculating the total cost of items when the price increases by a fixed amount each year due to inflation. Knowing the number of terms allows you to determine the total expenditure over a specific period.
