Calculating the Number of Terms in a Series: Solving for the Last Term and Options A, B, C, and D
Understanding the concept of series in mathematics is crucial for solving complex problems. A series is a sequence of numbers where each term after the first is obtained by adding a constant difference to the preceding term. This constant difference is also known as the common difference. In this article, we will delve into the process of calculating the number of terms in a series, specifically when the last term is known. We will use the series 5, 7, 9, …, 43 as an example and explore the options A – 14, B – 16, C – 20, and D – 19.
Understanding the Arithmetic Series
An arithmetic series is a sequence of numbers in which the difference of any two successive members is a constant. This constant is often referred to as the common difference. In the series 5, 7, 9, …, 43, the common difference is 2 (7-5).
Formula for the Last Term
The formula for the nth term of an arithmetic series is given by a + (n-1)d, where ‘a’ is the first term, ‘d’ is the common difference, and ‘n’ is the number of terms. In our case, a = 5, d = 2, and the last term is 43.
Calculating the Number of Terms
To find the number of terms in the series, we can rearrange the formula for the nth term to solve for ‘n’. This gives us n = [(Last Term – a)/d] + 1. Substituting the given values, we get n = [(43 – 5)/2] + 1 = 20. Therefore, the number of terms in the series is 20.
Exploring the Options
- Option A – 14: This option suggests that there are 14 terms in the series. However, using the formula, we calculated that there are 20 terms in the series. Therefore, Option A is incorrect.
- Option B – 16: Similarly, Option B is also incorrect as it does not match with our calculated number of terms.
- Option C – 20: This option matches with our calculated number of terms. Therefore, Option C is correct.
- Option D – 19: This option suggests that there are 19 terms in the series. However, we calculated that there are 20 terms in the series. Therefore, Option D is incorrect.
In conclusion, understanding the concept of series and the formula for the nth term of an arithmetic series is crucial for solving problems related to series. In this case, the correct option is C – 20, which matches with our calculated number of terms in the series.
