Arithmetic Series Solve Common Difference And Sum Of 500 Terms

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Understanding Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference, often denoted as 'd'. The first term of the series is usually denoted as 'a', and the nth term can be found using the formula:

a_n = a + (n - 1)d

where:

  • a_n is the nth term,
  • a is the first term,
  • n is the term number,
  • d is the common difference.

Furthermore, the sum of the first 'n' terms of an arithmetic series, denoted as S_n, can be calculated using the formula:

S_n = n/2 * [2a + (n - 1)d]

Alternatively, if the first term (a) and the last term (a_n) are known, the sum can also be calculated using:

S_n = n/2 * (a + a_n)

These formulas are essential tools for solving problems related to arithmetic series. Now, let's apply these concepts to the problem at hand.

Problem Statement

We are given the following information about an arithmetic series:

  • The first term (a) is 16.
  • The 21st term (a_21) is 24.

We are tasked with finding:

(a) The common difference (d) of the series.

(b) The sum of the first 500 terms (S_500) of the series.

This problem requires us to utilize the formulas mentioned earlier to determine the unknown values. We'll start by finding the common difference using the information about the first and 21st terms.

(a) Finding the Common Difference

To find the common difference, we'll use the formula for the nth term of an arithmetic series:

a_n = a + (n - 1)d

We know that the first term (a) is 16 and the 21st term (a_21) is 24. So, we can substitute these values, along with n = 21, into the formula:

24 = 16 + (21 - 1)d

Now, we solve for d:

24 = 16 + 20d

Subtract 16 from both sides:

8 = 20d

Divide both sides by 20:

d = 8/20

Simplify the fraction:

d = 2/5

Therefore, the common difference of the arithmetic series is 2/5 or 0.4. Understanding the common difference is crucial because it dictates how the series progresses. Each subsequent term is obtained by adding the common difference to the previous term. This constant addition is what defines the arithmetic progression. With the common difference calculated, we can now move on to the next part of the problem: finding the sum of the first 500 terms.

(b) Finding the Sum of the First 500 Terms

To find the sum of the first 500 terms, we'll use the formula for the sum of an arithmetic series:

S_n = n/2 * [2a + (n - 1)d]

We know the following values:

  • n = 500 (the number of terms)
  • a = 16 (the first term)
  • d = 2/5 (the common difference)

Substitute these values into the formula:

S_500 = 500/2 * [2(16) + (500 - 1)(2/5)]

Simplify the expression:

S_500 = 250 * [32 + (499)(2/5)]

S_500 = 250 * [32 + 998/5]

S_500 = 250 * [32 + 199.6]

S_500 = 250 * 231.6

S_500 = 57900

Therefore, the sum of the first 500 terms of the arithmetic series is 57900. This result highlights the power of arithmetic series formulas in efficiently calculating sums, especially when dealing with a large number of terms. The formula allows us to bypass the tedious process of adding each term individually, providing a direct and accurate solution. The sum represents the cumulative value of the series up to the 500th term, demonstrating the overall growth pattern of the arithmetic progression.

Conclusion

In this article, we successfully solved an arithmetic series problem by first finding the common difference and then calculating the sum of the first 500 terms. We utilized the formulas for the nth term and the sum of an arithmetic series, demonstrating their practical application. Understanding arithmetic series is crucial for various mathematical and real-world applications, including financial calculations, physics problems, and computer science algorithms. This step-by-step solution provides a clear methodology for approaching similar problems. By breaking down the problem into smaller parts and applying the appropriate formulas, we can efficiently arrive at the correct solution. Remember, the key to success in mathematics lies in understanding the underlying concepts and practicing problem-solving techniques. This article serves as a valuable resource for mastering arithmetic series and enhancing your mathematical skills.