Finding The 23rd Term And Sum Of First 23 Terms In An Arithmetic Sequence

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Delving into the realm of mathematical sequences, we encounter the fascinating world of arithmetic progressions. These sequences, characterized by a constant difference between consecutive terms, hold a unique charm and a wealth of practical applications. In this article, we embark on a journey to unravel the intricacies of a specific arithmetic sequence: 3, 5, 11, 17, 23. Our mission is twofold: first, to pinpoint the elusive 23rd term, and second, to calculate the grand total of the first 23 terms. Join us as we navigate the elegant formulas and logical steps that will lead us to these solutions.

Identifying the Arithmetic Sequence

Before we embark on our calculations, let's first confirm that the given sequence, 3, 5, 11, 17, 23, is indeed an arithmetic sequence. The hallmark of an arithmetic sequence is the constant difference between any two consecutive terms. To verify this, let's examine the differences between the terms:

  • 5 - 3 = 2
  • 11 - 5 = 6
  • 17 - 11 = 6
  • 23 - 17 = 6

Upon closer inspection, we observe that the difference between the first two terms (5 - 3 = 2) deviates from the consistent difference of 6 that prevails between the subsequent terms. This discrepancy leads us to the conclusion that the given sequence, 3, 5, 11, 17, 23, is not an arithmetic sequence in its entirety. The presence of the initial terms 3 and 5 disrupts the constant difference pattern that defines an arithmetic progression.

To proceed with our analysis and calculations, we must make a crucial adjustment. We will focus on the portion of the sequence that exhibits the arithmetic sequence property, which starts from the third term onwards. Therefore, we will consider the arithmetic sequence as: 11, 17, 23, ... This revised sequence maintains a constant difference of 6 between consecutive terms, making it a valid arithmetic progression.

Now that we have identified the arithmetic sequence, we can proceed to determine the 23rd term and the sum of the first 23 terms, using the appropriate formulas and techniques.

Determining the 23rd Term (a_23)

To find the 23rd term (a_23) of the arithmetic sequence, we will employ the fundamental formula that governs the behavior of arithmetic progressions. This formula provides a direct route to calculating any term in the sequence, given the first term, the common difference, and the term's position in the sequence. The formula is expressed as follows:

a_n = a_1 + (n - 1)d

Where:

  • a_n represents the nth term in the sequence (the term we want to find)
  • a_1 represents the first term of the sequence
  • n represents the position of the term we want to find (in this case, 23)
  • d represents the common difference between consecutive terms

In our specific arithmetic sequence, 11, 17, 23, ..., we have:

  • a_1 = 11 (the first term)
  • n = 23 (the position of the term we want to find)
  • d = 6 (the common difference)

Now, we can substitute these values into the formula to calculate a_23:

a_23 = 11 + (23 - 1) * 6 a_23 = 11 + (22) * 6 a_23 = 11 + 132 a_23 = 143

Therefore, the 23rd term (a_23) of the arithmetic sequence is 143. This means that if we were to continue the sequence, adding 6 to each term, the 23rd number in the sequence would be 143. This formula provides a powerful tool for extending arithmetic sequences and determining specific terms without having to manually calculate each term along the way.

Calculating the Sum of the First 23 Terms (S_23)

Having successfully determined the 23rd term of our arithmetic sequence, we now turn our attention to calculating the sum of the first 23 terms (S_23). This involves adding up all the terms from the first term (a_1) to the 23rd term (a_23). While we could manually add these terms, a more efficient and elegant approach is to use the formula specifically designed for calculating the sum of an arithmetic series. This formula streamlines the process and provides a direct path to the solution.

The formula for the sum of the first n terms of an arithmetic series is:

S_n = n/2 * (a_1 + a_n)

Where:

  • S_n represents the sum of the first n terms
  • n represents the number of terms (in this case, 23)
  • a_1 represents the first term of the sequence
  • a_n represents the nth term of the sequence (in this case, a_23, which we already calculated)

In our specific arithmetic sequence, we have:

  • n = 23 (the number of terms)
  • a_1 = 11 (the first term)
  • a_23 = 143 (the 23rd term, which we calculated in the previous section)

Now, we can substitute these values into the formula to calculate S_23:

S_23 = 23/2 * (11 + 143) S_23 = 23/2 * (154) S_23 = 23 * 77 S_23 = 1771

Therefore, the sum of the first 23 terms (S_23) of the arithmetic sequence is 1771. This value represents the total when we add up the first 23 numbers in the sequence, starting from 11 and increasing by 6 each time. This formula provides a concise and powerful method for calculating the sum of arithmetic series, saving us from the tedious task of manual addition.

Conclusion

In this exploration of the arithmetic sequence 11, 17, 23, ..., we have successfully navigated the realms of arithmetic progressions to determine both the 23rd term (a_23) and the sum of the first 23 terms (S_23). By applying the fundamental formulas that govern these sequences, we have arrived at the solutions:

  • The 23rd term (a_23) is 143.
  • The sum of the first 23 terms (S_23) is 1771.

These calculations showcase the elegance and efficiency of mathematical formulas in unraveling the patterns and properties of sequences. Arithmetic sequences, with their consistent differences and predictable behavior, serve as building blocks in various mathematical and real-world applications. From financial calculations to physics problems, understanding arithmetic progressions empowers us to analyze and model phenomena with precision.

This journey into the world of arithmetic sequences underscores the power of mathematical tools in revealing the hidden order and structure within numerical patterns. As we continue to explore the vast landscape of mathematics, we will undoubtedly encounter more fascinating sequences and series, each with its unique properties and applications. The quest to understand these mathematical constructs not only enhances our problem-solving abilities but also enriches our appreciation for the beauty and elegance of the mathematical universe.