Problem 65 - Beginning and Intermediate Algebra

Question

Evaluate each sum using a formula for \(S_{n}\). $$\sum_{i=1}^{7}(-2 i+7)$$

Step-by-Step Solution

Answer

The sum of the arithmetic series \(\sum_{i=1}^{7}(-2 i+7)\) can be found using the formula \(S_n = \frac{n}{2}(a + a_n)\), where \(a\) is the first term, \(a_n\) is the nth term, and \(n\) is the total number of terms. In this case, \(a = 5\), \(a_n = -7\), and \(n = 7\). Thus, \(S_7 = \frac{7}{2}(5 - 7) = -7\).

Step 1: 1. Identify the first term (a) and the common difference (d)

In the given arithmetic series, the first term is when the index i is equal to 1. So, a = -2(1) + 7 = 5 The common difference (d) is the factor that each consecutive term is differing from the last term. In this case, we can see that it is -2.

Step 2: 2. Calculate the nth term of the series (a_n)

As we have an arithmetic progression, we need the value of the nth term to continue. The formula for the nth term (a_n) in an arithmetic series is: \(a_n = a + (n-1)d\) Here, n is the total number of terms in the series, which is 7 in this problem. Substituting the known values in the nth term formula, we get: \(a_7 = 5 + (7-1)(-2)\) \(a_7 = 5 - 12 = -7\)

Step 3: 3. Use the formula for the sum of the first n terms (\(S_n\))

Now that we have the values of a, a_n, and n, we can use the formula for the sum of the first n terms of an arithmetic series: $$S_n = \frac{n}{2}(a + a_n)$$ Substituting the known values in the formula, we get: $$S_7 = \frac{7}{2}(5 - 7)$$

Step 4: 4. Calculate the sum

Solve the expression for the sum: $$S_7 = \frac{7}{2}(-2) = -7$$ So, the sum of the given arithmetic series is -7.

Key Concepts

Arithmetic Progression

An arithmetic progression is a sequence of numbers where the difference between any two consecutive terms is always the same. This difference is known as the "common difference." For example, in the sequence 3, 6, 9, 12, the common difference is 3.
The sequence can either increase or decrease:

• An increasing arithmetic progression has positive common difference.
• A decreasing arithmetic progression has negative common difference.

Recognizing an arithmetic progression can make the calculations for finding sums and other properties much easier.

nth term formula

In an arithmetic progression, the nth term formula allows us to find any term in the sequence without listing all terms up to that point. The formula is given by: \( a_n = a + (n - 1) d \) where:

• \(a\) is the first term of the sequence.
• \(d\) is the common difference.
• \(n\) is the position of the term in the sequence.

For instance, if the first term \(a\) is 5 and the common difference \(d\) is \(-2\), the 7th term \(a_7\) can be calculated as: \(a_7 = 5 + (7 - 1)(-2) = -7\).
This formula is useful because it gives a quick way to find any specific term without needing to sum through the whole sequence.

Sum formula for series

The sum of an arithmetic series involves adding up all the terms of an arithmetic progression. The formula for finding the sum \(S_n\) of the first \(n\) terms is: \[ S_n = \frac{n}{2} (a + a_n) \] where:

• \(n\) is the number of terms.
• \(a\) is the first term.
• \(a_n\) is the nth term.

This formula is derived by observing that pairing terms from the beginning and end of the sequence yields the same sum each time.
For example, using our previous values with \(a = 5\), \(a_7 = -7\), and \(n = 7\): \[ S_7 = \frac{7}{2} (5 + (-7)) = -7 \] This method saves time particularly with large sequences by avoiding long additions.

Common Difference

The common difference is a key component in understanding arithmetic progressions. It refers to the value that separates each pair of consecutive terms in a sequence. The common difference is consistent throughout the entire progression.
To find the common difference \(d\), subtract any term from the subsequent term in the sequence: \(d = a_2 - a_1\) or \(d = a_{i+1} - a_i\) for any consecutive terms.

• A positive common difference indicates that the sequence is increasing.
• A negative common difference indicates that the sequence is decreasing.
• A zero common difference means all terms are identical.

In our example, the common difference \(-2\) specifies that each term in this progression is 2 units less than the previous term. Hence, understanding the common difference helps in predicting and calculating terms in the sequence effortlessly.