Answer each question about the following arithmetic sequence:
[tex]$-20,-16,-12,-8,-4, \ldots$[/tex]
What is the 30th term of the sequence?
[tex]$a_{30}= $[/tex]
Answer (2)
Identify the first term a 1 = − 20 and the common difference d = 4 .
Apply the formula for the nth term of an arithmetic sequence: a n = a 1 + ( n − 1 ) d .
Substitute n = 30 , a 1 = − 20 , and d = 4 into the formula: a 30 = − 20 + ( 30 − 1 ) ( 4 ) .
Calculate the 30th term: a 30 = 96 . The final answer is 96 .
Explanation
Identifying the Sequence We are given an arithmetic sequence and asked to find the 30th term. Let's first identify the key components of the sequence.
Finding the Common Difference The first term of the sequence is a 1 = − 20 . The common difference, d , is the difference between consecutive terms. We can calculate it as d = − 16 − ( − 20 ) = 4 .
Using the Arithmetic Sequence Formula Now, we'll use the formula for the n th term of an arithmetic sequence, which is given by: a n = a 1 + ( n − 1 ) d
Calculating the 30th Term We want to find the 30th term, so n = 30 . Substituting the values a 1 = − 20 , d = 4 , and n = 30 into the formula, we get: a 30 = − 20 + ( 30 − 1 ) ( 4 ) a 30 = − 20 + ( 29 ) ( 4 ) a 30 = − 20 + 116 a 30 = 96
Final Answer Therefore, the 30th term of the arithmetic sequence is 96.
Examples
Arithmetic sequences are useful in many real-life situations. For example, imagine you save $100 each month. This forms an arithmetic sequence where the first term is $100 and the common difference is $100. After a year, you can easily calculate your total savings using the arithmetic sequence formula. Similarly, if a taxi charges a fixed initial fee plus a rate per mile, the total cost can be modeled using an arithmetic sequence. Understanding arithmetic sequences helps in predicting future values based on a consistent pattern.
The 30th term of the arithmetic sequence is 96, calculated using the formula for the nth term of an arithmetic sequence. We identified the first term as -20 and the common difference as 4, then applied these values in the formula. After calculation, we found that a 30 = 96 .
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