Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers such that the difference of any two successive members is a constant, known as the common difference. This concept is fundamental in mathematics and has wide applications in various fields such as finance, engineering, and computer science. This article will guide you through the process of finding the number of terms in an arithmetic sequence when given the first term, the last term, and the common difference.

Solving for Number of Terms in an Arithmetic Sequence

To find the number of terms (n) in an arithmetic sequence, we use the formula:

tn a (n - 1)d

Where:

tn is the nth term in the sequence. a is the first term in the sequence. n is the number of terms in the sequence. d is the constant difference between terms.

Let's apply this formula to the given problem, which involves the sequence: 7, 10, 13, ..., 55.

Step-by-Step Solution

Identify the first term (a) and the last term (tn). In this sequence, the first term (a) is 7, and the last term (tn) is 55. Determine the common difference (d). By subtracting the first term from the second term, we get: d term2 - term1 10 - 7 3 The common difference is 3, as it can be confirmed by: d term3 - term2 13 - 10 3. Substitute the values into the formula and solve for n:

55  7   (n - 1) * 3

Rearrange the equation to isolate n:

55 - 7  (n - 1) * 348  (n - 1) * 316  n - 1n  17

Therefore, the sequence has 17 terms.

Alternative Methods

There are other methods to find the number of terms in an arithmetic sequence, all leading to the same result. These methods involve:

Using the sum formula: If you are given the sum (S) of the series, you can use the formula S n/2 * (a l), where l is the last term. Solving for the nth term directly: If you know the nth term (an) and want to find n, you can rearrange the formula an a (n - 1)d to solve for n.

Additional Examples

Let's consider another example to solidify your understanding:

Given: an 43, a 7, and d 3.

Solve for n:

43  7   (n - 1) * 343 - 7  (n - 1) * 336  (n - 1) * 312  n - 1n  13

Thus, in this sequence, there are 13 terms.

Conclusion

Mastering the art of solving arithmetic sequences is essential for anyone looking to excel in mathematics or related fields. By understanding the basic formula and practicing with examples, you can confidently tackle similar problems. Keep practicing and exploring to deepen your understanding.