Discovering the 38th Term of the Arithmetic Sequence -9, -2, 5, 12, 19

When dealing with arithmetic sequences, it is important to understand the underlying pattern and apply the appropriate formulas to find specific terms. In this article, we will explore a detailed solution for finding the 38th term of the given sequence and discuss the general process for solving similar problems.

Understanding the Sequence

The given arithmetic sequence is -9, -2, 5, 12, 19. To find the common difference, we subtract any term from the one that follows it. Let's examine the differences between consecutive terms:

Common difference between -2 and -9: (-2 - (-9) 7) Common difference between 5 and -2: (5 - (-2) 7) Common difference between 12 and 5: (12 - 5 7) Common difference between 19 and 12: (19 - 12 7)

The common difference is consistently 7. This tells us that each term in the sequence is 7 more than the previous term.

Using the Arithmetic Sequence Formula

Arithmetic sequences can be defined by the formula:

nth term first term (n - 1) * common difference

For our specific problem, we need to find the 38th term. Here, the first term ((a_1)) is -9, the common difference ((d)) is 7, and (n) is 38.

We can plug these values into the formula:

an a1 (n - 1) * d

a38 -9 (38 - 1) * 7

To solve this, we first simplify the expression inside the parentheses:

38 - 1 37

Then multiply 37 by the common difference 7:

37 * 7 259

Finally, add this result to the first term -9:

-9 259 250

Therefore, the 38th term of the arithmetic sequence is 250.

Alternative Methods to Find the 38th Term

Let’s explore another way to find the 38th term by examining the pattern in a different manner.

Since we are adding 7 to each term, we can also consider multiplying the common difference by the total number of terms between -9 and 19 and adding the result to the last given term. This involves a bit more calculation:

If we multiply the common difference 7 by 37 (the number of terms from -9 to 19, inclusive) and add it to the last term 19:

19 (37 * 7) 19 259 278

Alternatively, we can use the formula with 38 terms:

-9 (38 - 1) * 7 -9 259 250

Both methods should yield the same term, confirming our previous calculation.

Conclusion and Further Exploration

Understanding and applying arithmetic sequence formulas is a fundamental skill in mathematics. Whether you are working with real-world sequences or solving abstract problems, the ability to identify patterns and use the appropriate formula is crucial.

To further explore this topic, you can practice similar problems and delve deeper into more complex sequences and series, such as geometric sequences or sequences defined by recursive formulas. By mastering these concepts, you will be well-equipped to tackle a wide range of mathematical challenges.