Problem 66 - Precalculus with Limits

Question

In Exercises 59 - 66, find the indicated \( n \)th partial sum of the arithmetic sequence. \( a_1 = 15, a_{100} = 307, n = 100 \)

Step-by-Step Solution

Answer

The 100th partial sum of the given arithmetic sequence is 16100

Step 1: Identify known values

From the question, we can identify the known values. Here, \(a_1 = 15\), \(a_{100} = 307\), and \(n = 100\).

Step 2: Substitute into formula for nth partial sum

As per the formula \(S_n = n/2 * (a_1 + a_n)\), we substitute the known values into the formula: \(S_{100} = 100/2 * (15 + 307)\).

Step 3: Simplify the equation

In this step, we will perform the operations as per BODMAS rule (Bracket, Of, Division, Multiplication, Addition, and Subtraction). Here, sum of first and last terms is \(15 + 307 = 322\), then division by 2 gives \(100/2=50\). On multiplication of these results, we get \(50 * 322 = 16100\).

Step 4: Conclusion

After all the calculations, \(S_{100} = 16100\). This means that the sum of the first 100 terms of the arithmetic sequence is 16100.

Key Concepts

Partial Sum

To understand the concept of the partial sum in an arithmetic sequence, start by considering what a sequence is. An arithmetic sequence is a list of numbers where each term after the first is obtained by adding a constant difference to the previous term. The partial sum, denoted as \( S_n \), is the sum of the first \( n \) terms of this sequence.

The formula used to calculate the partial sum in an arithmetic sequence is:

• \( S_n = \frac{n}{2} (a_1 + a_n) \)

where \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the \( n\)th term. This formula simplifies the process by using the average of the first and the last terms and multiplying it by the number of terms.

In our example, \( S_{100} = \frac{100}{2} (15 + 307) \) results in 16100, illustrating how partial sums make it straightforward to calculate the total sum of a specified number of terms.

Nth Term

The n-th term of an arithmetic sequence represents any term located at a position \( n \) in the sequence. Each term is computed from the first term and the common difference \( d \), according to the formula:

• \( a_n = a_1 + (n-1) \, d \)

Here, \( a_n \) is the \( n \)th term, and \( a_1 \) is the first term.

Understanding this formula helps in locating any specific term in the sequence without needing to know all preceding terms. In the given problem, we're told that \( a_{100} = 307 \), showing how this term fits within the sequence. Using these portions of information helps verify the structure and correctness of the arithmetic sequence as a whole.

Sequence Formula

The sequence formula is fundamental for arithmetic sequences as it guides how each succeeding term is formed. This formula, \( a_n = a_1 + (n-1) \times d \), is essential for discovering the value of any term in an arithmetic sequence.

Here:

• \( a_n \) represents the \( n \)th term.
• \( a_1 \) is the first term of the sequence.
• \( d \) is the constant difference between consecutive terms.
• \( n \) is the position of the term in the sequence.

Applying this formula helps to predict and understand the formation of the sequence. For example, with \( a_1 = 15 \) and \( a_{100} = 307 \), you can infer other terms by manipulating \( d \), the common difference. This understanding reinforces your grasp of how sequences expand and how to verify any part of them efficiently.