Calculating Terms in an Arithmetic Progression: A Comprehensive Guide

Understanding arithmetic progressions (AP) is a fundamental concept in mathematics. An arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the preceding term. This article will guide you through the process of finding the number of terms in a given arithmetic progression, specifically focusing on the sequence 7, 10, 13, ..., 43.

Understanding the Formula for the nth Term

The formula to find the n-th term of an arithmetic progression can be expressed as: $$a_n a (n-1)d$$ where: - a is the first term of the progression, - d is the common difference, - n is the number of terms, - a_n is the n-th term of the sequence.

Solving the Given Sequence

Our sequence is 7, 10, 13, ..., 43. We need to determine the number of terms in this sequence.

Step 1: Identify the first term, a, and the common difference, d. - a 7 - d 10 - 7 3

Step 2: Apply the formula to the last term of the sequence, which is a_n 43.

$$a_n a (n-1)d$$ $$43 7 (n-1)3$$

Step 3: Solve for n.

Subtract 7 from both sides:

$$43 - 7 (n-1)3$$ $$36 (n-1)3$$

Divide both sides by 3:

$$n - 1 12$$

Add 1 to both sides:

$$n 13$$

Therefore, there are 13 terms in the arithmetic progression 7, 10, 13, ..., 43.

Verification with a Table

Let's verify the result with a table of terms to ensure accuracy:

Term (n) Value (a_n) 1 7 2 10 3 13 13 43

Additional Insights

It's interesting to note that the sequence between 13 and 43 also follows the same arithmetic progression. Specifically, starting from 13 and ending at 43, there are 8 terms:

13 16 19 22 25 28 31 34 37 40 43

To verify, the difference between each term is 3, and the calculation confirms 8 terms.

Conclusion

By using the formula for the n-th term of an arithmetic progression, we can accurately determine the number of terms in a sequence. In the case of the sequence 7, 10, 13, ..., 43, there are 13 terms in total, and 8 terms between 13 and 43.

Related Keywords

This article covers topics such as 'arithmetic progression' and 'common difference', which are key concepts in understanding sequences and series in mathematics.

References

This content is based on mathematical principles and is verified using the given arithmetic progression formula.