A series is given below:
[tex]$7+11+15+19$[/tex]
a) What type of series is it?
b) Find [tex]$n^{t_1}$[/tex] term of series.
c) Express the series in sigma notation.
Answer (1)
The series is identified as an arithmetic series because the difference between consecutive terms is constant.
The n t h term of the arithmetic series is found using the formula a n = a + ( n − 1 ) d , resulting in a n = 4 n + 3 .
The series is expressed in sigma notation as ∑ n = 1 4 ( 4 n + 3 ) .
The type of series, n t h term, and sigma notation are determined, and the final answer is n = 1 ∑ 4 ( 4 n + 3 ) .
Explanation
Problem Analysis The given series is 7 + 11 + 15 + 19 + ... . We need to determine the type of series, find the n t h term, and express the series in sigma notation.
Identifying the Series Type To identify the type of series, we calculate the difference between consecutive terms: 11 − 7 = 4 , 15 − 11 = 4 , 19 − 15 = 4 . Since the difference between consecutive terms is constant, the series is an arithmetic series.
Finding the nth Term For an arithmetic series, the n t h term is given by a n = a + ( n − 1 ) d , where a is the first term and d is the common difference. In this series, the first term a = 7 and the common difference d = 4 . Therefore, the n t h term is a n = 7 + ( n − 1 ) 4 = 7 + 4 n − 4 = 4 n + 3 .
Expressing in Sigma Notation To express the series in sigma notation, we use the general term 4 n + 3 . The series starts with n = 1 . So, the sigma notation is ∑ n = 1 N ( 4 n + 3 ) , where N is the number of terms in the series. Since the given series is 7 + 11 + 15 + 19 , it has 4 terms. Thus, the sigma notation is ∑ n = 1 4 ( 4 n + 3 ) .
Final Answer a) The series is an arithmetic series. b) The n t h term of the series is 4 n + 3 .
c) The series in sigma notation is ∑ n = 1 4 ( 4 n + 3 ) .
Examples
Arithmetic series are useful in various real-life scenarios, such as calculating simple interest, predicting the cost of items with a fixed annual increase, or determining the number of seats in an auditorium where each row has a fixed number of additional seats compared to the previous row. For example, if you deposit $100 into a savings account that earns $5 simple interest each year, the total amount in your account each year forms an arithmetic series: $105, $110, $115, and so on. Understanding arithmetic series helps in financial planning and forecasting.
