Find the value of the 11th term of the arithmetic sequence: -2, 5, 12, 19, 26, ... using the formula [tex]a_n=a_1+(n-1) d[/tex].
Answer (1)
Identify the first term a 1 = − 2 and the common difference d = 7 .
Substitute a 1 = − 2 , n = 11 , and d = 7 into the formula a n = a 1 + ( n − 1 ) d .
Calculate a 11 = − 2 + ( 11 − 1 ) ( 7 ) = − 2 + ( 10 ) ( 7 ) = − 2 + 70 .
Simplify to find the 11th term: a 11 = 68 . The final answer is 68 .
Explanation
Understanding the Problem We are given an arithmetic sequence and asked to find the 11th term. The formula for the nth term of an arithmetic sequence is given by a n = a 1 + ( n − 1 ) d , where a 1 is the first term, n is the term number, and d is the common difference.
Identifying the First Term and Common Difference First, we need to identify the first term ( a 1 ) and the common difference ( d ) of the given sequence: − 2 , 5 , 12 , 19 , 26 , … The first term is a 1 = − 2 .
The common difference is the difference between consecutive terms, which is d = 5 − ( − 2 ) = 7 .
Substituting the Values Now, we can substitute the values a 1 = − 2 , n = 11 , and d = 7 into the formula for the nth term: a 11 = a 1 + ( n − 1 ) d = − 2 + ( 11 − 1 ) ( 7 )
Calculating the 11th Term Next, we simplify the expression: a 11 = − 2 + ( 10 ) ( 7 ) = − 2 + 70 = 68
Final Answer Therefore, the 11th term of the arithmetic sequence is 68.
Examples
Arithmetic sequences are useful in many real-life situations, such as calculating loan payments or predicting the growth of a population. For example, if you deposit a fixed amount of money into a savings account each month, the total amount in the account over time forms an arithmetic sequence. Understanding how to calculate terms in an arithmetic sequence can help you plan your savings and investments effectively. Let's say you deposit $100 each month into an account that doesn't earn interest. After the first month, you have $100. After the second month, you have $200, and so on. The amount in your account each month forms an arithmetic sequence with a common difference of $100.
