Problem 64 - Beginning and Intermediate Algebra

Question

Evaluate each sum using a formula for \(S_{n}\). $$\sum_{i=1}^{9}(2 i-14)$$

Step-by-Step Solution

Answer

The sum of the arithmetic series \(\sum_{i=1}^{9}(2 i-14)\) can be evaluated using the formula \(S_n = \dfrac{n}{2} (a_1 + a_n)\) where \(n=9\), \(a_1=-12\), and \(a_9=4\). Applying the formula, we get \(S_9 = -36\).

Step 1: Identifying the arithmetic sequence

The given summation is \(\sum_{i=1}^{9}(2 i-14)\), which stands for the arithmetic sequence \(2(1) - 14, 2(2) - 14, 2(3) - 14, \ldots, 2(9) - 14\).

Step 2: Find the first term

Plug in the index \(i=1\) into the sequence formula to find the first term: \(a_1 = 2(1) - 14 = -12\).

Step 3: Find the last term

Plug in the index \(i=9\) into the sequence formula to find the last term: \(a_9 = 2(9) - 14 = 4\).

Step 4: Obtain the number of terms

The index goes from 1 to 9, meaning we have 9 terms in total.

Step 5: Use formula to find the sum

Apply the formula for the sum of the first n terms of an arithmetic series: \(S_n = \dfrac{n}{2} (a_1 + a_n)\) \(S_9 = \dfrac{9}{2} (-12 + 4)\) \(S_9 = \dfrac{9}{2} (-8)\) \(S_9 = -36\) So, the sum of the given arithmetic series is -36.

Key Concepts

Sum of Arithmetic Series

When dealing with arithmetic series, we often aim to find the sum of all the terms. This sum is found using a specific formula geared towards arithmetic sequences. Arithmetic series are simply the sum of terms in an arithmetic sequence. These sequences have a common difference between each consecutive term.

The sum of the first \(n\) terms of an arithmetic series is given by:

• \(S_n = \dfrac{n}{2} (a_1 + a_n)\)

This formula is extremely helpful because instead of adding all numbers individually, we can simply use the first and last term to find the total sum.
Remember, \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term in the sequence.

First Term of Sequence

The first term of an arithmetic sequence is often the starting point for understanding the series itself. To find the first term, we substitute the initial index of the sequence into the given formula.

In our exercise, the formula \(2i - 14\) is used, where the first index \(i = 1\). So, we calculate:

• \(a_1 = 2(1) - 14 = -12\)

The first term \(-12\) tells us a lot. It's not only where our sequence begins, but it also impacts the common difference and the progression of all further terms in this arithmetic sequence.

Last Term of Sequence

The last term in an arithmetic sequence helps us understand where our sequence concludes. It’s crucial when applying the sum of sequence formula. To determine this, substitute the final index into the sequence formula.

For example, using the formula \(2i - 14\), and the last index \(i = 9\), we find:

• \(a_9 = 2(9) - 14 = 4\)

Knowing the last term \(4\) helps complete the arithmetic series. It clarifies the entire range of numbers you're working with and is essential for calculating sums.

Number of Terms in Sequence

The number of terms in an arithmetic sequence tells us how many elements there are from start to finish. This figure is critical for using the sum formula, as it directly influences the total. To find this, simply determine the range of your index.

In our example, our index goes from \(i = 1\) to \(i = 9\). Therefore:

• The number of terms \(n = 9\)

Each term depends on the previous one by the common difference, making the sequence predictable and easily computable.

Sequence Formula

A sequence formula provides a rule or pattern for determining any term in an arithmetic sequence. It outlines how each term is derived based on the index \(i\).

For the arithmetic sequence in our exercise, the formula \(2i - 14\) tells us:

• For any term \(a_i\), calculate it by substituting \(i\) into \(2i - 14\).

This formula is the essence of our sequence, providing a straightforward method to find terms without listing them all. It defines both how our sequence starts and how it progresses with each subsequent term.