What is the 20th term of the arithmetic sequence [tex]3,9,15,21,27, \ldots ? \ a_{20}=\square[/tex]
Answer (2)
Identify the first term a 1 = 3 and the common difference d = 6 .
Use the formula for the nth term of an arithmetic sequence: a n = a 1 + ( n − 1 ) d .
Substitute n = 20 , a 1 = 3 , and d = 6 into the formula: a 20 = 3 + ( 20 − 1 ) × 6 .
Simplify the expression to find the value of a 20 : a 20 = 117 .
Explanation
Understanding the Problem We are given an arithmetic sequence and asked to find the 20th term. An arithmetic sequence is a sequence where the difference between consecutive terms is constant. In this case, the sequence is 3, 9, 15, 21, 27, ...
Identifying Key Information The first term of the sequence is a 1 = 3 . The common difference d is the difference between any two consecutive terms. For example, 9 − 3 = 6 , 15 − 9 = 6 , so d = 6 . We want to find the 20th term, which we denote as a 20 .
Recalling the Formula The formula for the n th term of an arithmetic sequence is given by: a n = a 1 + ( n − 1 ) d
Substituting Values We are looking for the 20th term, so n = 20 . We know that a 1 = 3 and d = 6 . Substituting these values into the formula, we get: a 20 = 3 + ( 20 − 1 ) "."6
Calculating the 20th Term Now, we simplify the expression: a 20 = 3 + ( 19 ) "."6 a 20 = 3 + 114 a 20 = 117
Final Answer Therefore, the 20th term of the arithmetic sequence is 117.
Examples
Arithmetic sequences are useful in many real-life situations. For example, if you save a fixed amount of money each month, the total amount you've saved over time forms an arithmetic sequence. Understanding arithmetic sequences can help you predict how much money you'll have saved after a certain number of months. Another example is calculating the number of seats in a stadium where each row has a fixed number of additional seats compared to the row before it.
The 20th term of the given arithmetic sequence is 117. This term was calculated using the first term and the common difference. The formula used is a n = a 1 + ( n − 1 ) d .
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