Problem 73 - Precalculus with Limits

Question

In Exercises 67 - 74, find the partial sum. \( \sum_{n=1}^{500}\left(n + 8\right) \)

Step-by-Step Solution

Answer

The partial sum of the series is \(129250\).

Step 1: Identify the summation formula

We first have to identify what the sum looks like. In this case, it's the sum of the terms where each term is given by the function \(n+8\), where n goes from 1 to 500. That is, we are adding up all the numbers from 1 to 500, and then adding 8 to each of these numbers before summing them up.

Step 2: Break down the sum into two parts

Now we break down the sum into two parts. The sum from 1 to 500 of \(n+8\) equals to the sum from 1 to 500 of \(n\) plus the sum from 1 to 500 of \(8\). This is a property of summations and can be written as: \(\sum_{n=1}^{500}\left(n + 8\right) = \sum_{n=1}^{500} n + \sum_{n=1}^{500} 8\)

Step 3: Calculate each sum separately

The first part, \(\sum_{n=1}^{500} n\), could be calculated using the formula of sum of an arithmetic series, which is \((numberOfTerms/2)*(firstTerm + lastTerm)\), results in \(500/2 * (1+500) = 125250\). The second part sum, \(\sum_{n=1}^{500} 8\), is easy to calculate, because every term is 8, so the sum is just \(8*500 = 4000\).

Step 4: Sum the two parts together

Now simply sum the two parts together. The total sum is \(125250 + 4000\) which equals to \(129250\).

Key Concepts

Arithmetic Series

An arithmetic series is a type of sequence in which each term is derived by adding a constant to the previous term. In our given problem, the sequence is defined by the function \( n+8 \), which involves both an arithmetic sequence and a constant shift. An arithmetic sequence like \( n \) (from 1 to 500) is where each number increases by 1.
To find the sum of such a sequence, it is crucial to understand a key formula:

• The sum of an arithmetic sequence is calculated as \(( \text{number of terms} / 2) \times (\text{first term} + \text{last term} )\).

For our exercise, when breaking down the problem, you can consider \( n \) as an arithmetic sequence and 8 as a constant value added to each term. This makes the sum consist of two distinct parts - the arithmetic growth (the \( n \) term) and the constant adjustment (the \(+8\)). Understanding this separation makes solving the sequence simpler and allows for clear calculations.

Summation Formula

The summation formula is a powerful tool for quickly calculating the sum of terms in a series without needing to add each term individually. This formula is especially handy in the context of arithmetic series.
The formula for the sum of the first \( n \) natural numbers, for instance, is given by:

• \( \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \)

In the exercise, the summation formula allows us to easily calculate \( \sum_{n=1}^{500} n \), which represents the sum of the first 500 natural numbers. Once you apply this formula, you can instantly determine such sums accurately, making the arithmetic handling of large numbers less daunting. By separating the \( n \) part and the constant part, the summation formula provides the prerequisite steps to effectively handle series calculations.

Series Calculation

Calculating a series involves determining the total from adding all the terms defined in the series's formula. In the presented problem, the formula for the series was \( n+8 \), ranging from \( n=1 \) to \( n=500 \).
We tackled this by breaking down the series into simpler components:

• First, compute \( \sum_{n=1}^{500} n \) using arithmetic series techniques, resulting in 125,250.
• Next, calculate \( \sum_{n=1}^{500} 8 \), which simplifies as the constant 8 multiplied by 500, yielding 4,000.

These calculations make it clear how each part contributes to the final sum. Adding these results together gives the total value of 129,250 for the series. Through this structured approach, even seemingly complex series become manageable to solve. It's all about understanding the individual parts and knowing how to work with them effectively.