In the sequence 3, 6, 9 ..., which term is 270?
Answer (2)
Identify the first term a = 3 and the common difference d = 3 .
Use the formula for the n -th term of an arithmetic sequence: a n = a + ( n − 1 ) d .
Substitute a n = 270 , a = 3 , and d = 3 into the formula: 270 = 3 + ( n − 1 ) 3 .
Solve for n : n = 90 . Thus, the 90th term is 270. 90
Explanation
Understanding the Problem We are given an arithmetic sequence (AS) 3, 6, 9, ... and we want to find which term in the sequence is equal to 270.
Identifying the First Term and Common Difference The first term of the sequence, denoted as a , is 3. The common difference, denoted as d , is the difference between consecutive terms, which is 6 − 3 = 3 .
Stating the General Formula The general formula for the n -th term of an arithmetic sequence is given by: a n = a + ( n − 1 ) d
Substituting the Given Values We are given that a n = 270 . We need to find the value of n for which this is true. Substituting the values of a , d , and a n into the formula, we get: 270 = 3 + ( n − 1 ) 3
Solving for n Now, we solve for n :
270 = 3 + 3 n − 3 270 = 3 n n = 3 270 n = 90
Final Answer Therefore, the 90th term of the arithmetic sequence is 270.
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 how to find a specific term in a sequence can help you predict how much 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 previous row.
The term 270 is the 90th term in the arithmetic sequence 3, 6, 9, ... This is found by using the formula for the n -th term of an arithmetic sequence. By calculating, we determine that n = 90 .
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