Problem 50 - College Algebra

Question

For Exercises \(45-50,\) write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. $$\sum_{i=1}^{50}(-4 i)$$

Step-by-Step Solution

Answer

The first three terms are -4, -8, -12, and the last term is -200. The sum from i=1 to 50 of \(-4i\) is -5100.

Step 1: Identify the first three terms and the last term

The sequence given in the function is \(-4i\), where 'i' is the position of the term in the sequence. So the first term is \(-4 \cdot 1 = -4\), the second term is \(-4 \cdot 2 = -8\) and the third term is \(-4 \cdot 3 = -12\). The fiftieth term, \(-4 \cdot 50 = -200\).

Step 2: Use formula of Sum of Arithmetic Sequence

Now that we have the first and last terms where the first term (a_1) is -4, the last term (a_n) is -200, and the total number of terms (n) is 50, we can input this into the summing formula for an arithmetic sequence which is \(S_n = n/2(a_1 + a_n)\)

Step 3: Compute the sum

Let's substitute the values into the formula: \(S_{50} = 50/2(-4+-200) = 25(-204) = -5100\)

Key Concepts

First Three Terms

In an arithmetic sequence, each term is obtained by adding a constant difference to the previous term. In this example, the sequence is defined by the function \(-4i\), where \(i\) is the position of the term starting from 1. To understand the sequence better, let's focus on the first three terms which are a critical starting point for any arithmetic sequence problem.To find the first term, substitute \(i = 1\) into the sequence formula: \(-4 \cdot 1 = -4\). This calculates the initial term of the series.
For the second term, plug in \(i = 2\): \(-4 \cdot 2 = -8\).
The third term requires \(i = 3\): \(-4 \cdot 3 = -12\).
This simple calculation helps us identify the initial behavior of the sequence, which ensures we can tackle further tasks such as calculating the sum or finding the last term.

Sum Formula

The sum of an arithmetic series, which consists of \(n\) terms, can be easily calculated using a special formula. Understanding this formula is key to solving sums effectively and efficiently.The formula for the sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is:\[S_n = \frac{n}{2} (a_1 + a_n)\]where:

• \(n\) is the total number of terms,
• \(a_1\) is the first term, and
• \(a_n\) is the last term.

Here, for the sequence defined, we have identified that there are 50 terms, with the first and last terms being \(-4\) and \(-200\) respectively. By plugging these values into the formula, you can compute the sum of the entire arithmetic sequence.

Arithmetic Sequence

An arithmetic sequence is a list of numbers with a fixed difference between each pair of consecutive terms. In our case, the sequence follows a straightforward pattern defined by \(-4i\).The difference between consecutive terms can be found using any two successive terms from the sequence: \(-8 - (-4) = -4\) or equally, \(-12 - (-8) = -4\). This confirms our common difference is \(-4\), which is consistent with the formula's coefficient.Understanding that each term in this sequence is generated by adding \(-4\) to the previous term is crucial. It helps not only in identifying individual terms but also in comprehending the overall structure of the sequence, facilitating further manipulations like calculating sums or predicting future terms.

Last Term

Determining the last term is often essential for various arithmetic sequence calculations, especially when working with the sum formula. The last term can indicate important insights into the sequence's range and behavior.In this exercise, the last term is calculated by applying the function \(-4i\) with \(i=50\): \(-4 \cdot 50 = -200\). This confirms that for our sequence, the final term after 50 sequences is \(-200\). Knowledge of both the first and last terms is fundamental for employing the sum formula that calculates the total of the sequence. By establishing these endpoints, you're equipped to fully explore and solve the arithmetic series problem at hand.