Finding the Nth Term of the Sequence 15, 12, 9, 6
When working with sequences, one common problem is to find the nth term. In this article, we will go through the process of determining the nth term for the sequence 15, 12, 9, 6. We will break down the steps and provide detailed explanations to ensure a clear understanding.
Understanding the Sequence
The given sequence is 15, 12, 9, 6. By examining the sequence, we can observe that it decreases by 3 each time. This indicates that it is an arithmetic sequence, where each term is obtained by adding or subtracting a constant (the common difference).
Determining the Nth Term
To find the nth term of an arithmetic sequence, we can use the formula for the nth term of an arithmetic sequence:
an a1 (n - 1)d
an - The nth term of the sequence. a1 - The first term of the sequence. d - The common difference. n - The term number.In this sequence, the first term (a_1 15) and the common difference (d -3). Substituting these values into the formula:
an 15 (n - 1) * (-3)
Simplifying the formula:
an 15 - 3n 3
an 18 - 3n
Therefore, the nth term of the sequence is given by:
an 18 - 3n
Verification
To ensure the correctness of the formula, let's verify it with the first few terms:
1st term: (18 - 3(1) 15) 2nd term: (18 - 3(2) 12) 3rd term: (18 - 3(3) 9) 4th term: (18 - 3(4) 6)The results match the given sequence, confirming that the formula is correct.
Conclusion
In summary, the nth term of the sequence 15, 12, 9, 6 is given by the formula:
an 18 - 3n
This formula can be used to find any term in the sequence by substituting the value of n.
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By thoroughly understanding the nth term and its application in arithmetic sequences, you can solve a wide range of problems efficiently.