Problem 30 - Algebra 2

Question

Determine whether each series is arithmetic or geometric. Then evaluate the finite series for the specified number of terms. \(1+2+3+4+\ldots ; n=1000\)

Step-by-Step Solution

Answer

The series is an arithmetic series, and the sum of the first 1000 terms is 500500.

Step 1: Determine the type of series

A series is arithmetic if the difference of consecutive terms is constant. A series is geometric if the ratio of consecutive terms is constant. Here, the difference between consecutive terms is constant (increments by 1 between terms), thus, it is an Arithmetic series.

Step 2: Find the nth term of the arithmetic series

For an arithmetic series, the nth term, \(a_n\), can be found by the formula \(a_n = a_1 + (n-1) * d\) where \(a_1\) is the first term and \(d\) is the common difference. The first term \(a_1 = 1\), and the common difference \(d = 1\). Substitute these into the formula, we get \(a_{1000} = 1 + (1000 - 1) * 1 = 1000\). So the 1000th term of the series is 1000.

Step 3: Calculate the sum of series up to 1000 terms

Next, we calculate the sum of the first n=1000 terms. There is a formula for the sum of an arithmetic series: \(S_n = \frac{n}{2}*(a_1 + a_n)\) where \(a_1\) is the first term and \(a_n\) is the nth term. Plugging in the values we have, \(S_{1000} = \frac{1000}{2} * (1 + 1000) = 500500\).

Key Concepts

Sum of Arithmetic Series

Understanding how to find the sum of an arithmetic series can be incredibly useful, especially when dealing with a large number of terms. In general, an arithmetic series is the sum of terms in a sequence with a constant difference between them.

The sum of an arithmetic series can be quickly and effectively calculated using the formula:

• Given as, \( S_n = \frac{n}{2} \times (a_1 + a_n) \)
• Where \( S_n \) is the sum of the series, \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the nth term.

This formula simplifies what would otherwise be a daunting task of adding each individual term manually, especially when \( n \) (like in our exercise) is as large as 1000. By substituting known values into the formula, you can find the sum of the series without too much hassle.

Arithmetic Series Formula

The formula for an arithmetic series is crucial because it helps us find specific terms in a sequence or the sum of terms. This formula takes advantage of the constant difference between terms.

For the nth term, the formula is:

• \( a_n = a_1 + (n-1) \times d \)
• Where \( a_n \) represents the nth term, \( a_1 \) is the first term, \( n \) is the position of the term in the sequence, and \( d \) is the common difference between terms.

Using this formula helps to calculate any term in the arithmetic sequence, not just the 1000th term as used in this exercise. In instances where you have a large number of terms, this formula proves to be especially efficient and saves time.

Finite Series Evaluation

Evaluating a finite arithmetic series means finding the total sum of a series that has a set number of terms. In our specific example from the exercise, it involves calculating the sum of the first 1000 terms of the series.

Finite series evaluation is facilitated by determining the series type and using the appropriate formula. Given our series is clearly arithmetic due to the constant difference of 1 between terms, we can confidently apply the arithmetic series sum formula. This process involves:

• Identifying the series type.
• Finding the nth term using the arithmetic series formula if it's not provided.
• Utilizing the sum formula for arithmetic series: \( S_n = \frac{n}{2} \times (a_1 + a_n) \).

This allows for a quick evaluation of the sum, ensuring that you can determine the total value of a finite series thoughtfully and accurately.