Finding the 36th Term in an Arithmetic Sequence: A Step-by-Step Guide
Arithmetic sequences, a fundamental topic in mathematics, are sequences of numbers in which the difference between any two consecutive terms is constant. This article will guide you through the process of finding the 36th term in the arithmetic sequence 10, 4, -2. We will break down the steps, explain the concept of a common difference, and demonstrate the formula for calculating the nth term in a sequence. By the end, you will be able to solve similar problems on your own.
Understanding the Sequence
An arithmetic sequence is defined by its first term and common difference. In our example, the sequence starts at 10, and the common difference, denoted by (d), is calculated by subtracting the first term from the second term. Let's determine the common difference:
Step 1: Determine the Common Difference
The common difference (d) can be found by subtracting the first term from the second term:
[d 4 - 10 -6]
Step 2: Use the Formula for the (n)-th Term
The (n)-th term of an arithmetic sequence can be found using the formula:
[t_n a (n-1)d]
Where (a) is the first term, (d) is the common difference, and (n) is the term number.
Step 3: Calculate the 36th Term
Now, let's calculate the 36th term ((t_{36})) of the sequence using the formula:
[t_{36} a (36-1)d]
Substitute (a10) and (d-6):
[t_{36} 10 (36-1)(-6)]
[t_{36} 10 35(-6)]
[t_{36} 10 - 210]
[t_{36} -200]
Conclusion
In conclusion, the 36th term in the arithmetic sequence 10, 4, -2 is -200. This process demonstrates the step-by-step calculation using the common difference and the formula for the (n)-th term. Understanding the concept and applying the formula can help solve similar problems involving arithmetic sequences.