Problem 72 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the partial sum without using a graphing utility. $$\sum_{n=1}^{100} 2 n$$

Step-by-Step Solution

Answer

The partial sum of the series is 10100.

Step 1: Identify number of terms

First identify the number of terms in the series. In this case, since the series goes from \(n=1\) to \(n=100\), there are \(100\) terms in the series.

Step 2: Find the first term 'a1' and last term 'an'

The first term of the series \(a_1\) is obtained by plugging \(n=1\) into the series equation, giving \(2 * 1 = 2\). The last term \(a_n\) is obtained by plugging \(n=100\) into the series equation, giving \(2 * 100 = 200\).

Step 3:Compute the Partial Sum

After identifying the number of terms, and the first and last terms, use the arithmetic series sum formula to calculate the partial sum. Here, plug \(n=100\), \(a_1 = 2\), \(a_100 = 200\) into the formula \(\sum_{i=1}^{n} = n/2 * (a_1 + a_n)\) to get the partial sum as \(\sum_{i=1}^{100} = 100/2 * (2 + 200) = 10100\).

Key Concepts

Partial Sum

The partial sum of an arithmetic sequence refers to the total sum of the first few terms in that sequence. Partial sums are useful when you need to find the sum without calculating each term individually. In the context of our problem, finding the partial sum means determining the total of all terms from the first term up to the specified term. The formula to calculate the partial sum of an arithmetic sequence is \( S_n = \frac{n}{2} (a_1 + a_n) \), where:

• \( S_n \) is the partial sum of the sequence.
• \( n \) is the number of terms to consider.
• \( a_1 \) is the first term.
• \( a_n \) is the nth term or the last term in the series.

This formula simplifies the process, as it only requires the first and last term along with the number of terms to find the sum. It's efficient because we avoid summing up each term one by one, especially in cases of long sequences.

Number of Terms

Understanding the number of terms in a sequence is crucial for calculating sums and analyzing arithmetic patterns. The number of terms is simply the count of all terms from the start of the sequence to the end of the specified part of the sequence. In arithmetic sequences, determining the number of terms is straightforward if the range is given. For example:

• If the series starts at \( n=1 \) up to \( n=100 \), there are exactly 100 terms.

Knowing the number of terms is the first critical step when you're working with the arithmetic series sum formula. This information plays a key role in managing calculations, especially in applications like finding the partial sum or estimating series length. It ensures you plug the right numbers into formulas, leading to correct results.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers where each term after the first is the result of adding a constant to the previous term. This constant is known as the "common difference." Arithmetic sequences are straightforward because their pattern can be easily identified and predicted. For example, in the sequence \( 2, 4, 6, 8, ... \), the common difference is 2.

• The general form of an arithmetic sequence is given by \( a_n = a_1 + (n-1)d \).
• \( a_n \) is the nth term.
• \( a_1 \) is the first term.
• \( d \) is the common difference.

By recognizing the arithmetic pattern, calculations like finding the nth term or the partial sum become much more manageable. Whether you're dealing with finance, computer science, or physics, understanding how to utilize arithmetic sequences is an essential skill across many fields.