Problem 66 - Beginning and Intermediate Algebra

Question

Evaluate each sum using a formula for \(S_{n}\). $$\sum_{i=1}^{12}(-4 i+9)$$

Step-by-Step Solution

Answer

Using the formula for the sum of an arithmetic sequence, \(S_n = \frac{n(a_1 + a_n)}{2}\), with \(n = 12\), \(a_1 = 5\), and \(a_n = -39\), we get: \(S_{12} = \frac{12(5 + (-39))}{2} = -204\). Therefore, \(\sum_{i=1}^{12}(-4 i+9) = -204\).

Step 1: Identify the number of terms, n

The given sum has an upper limit of 12, which means there are 12 terms in the sum. Therefore, \(n = 12\).

Step 2: Find the first term, \(a_1\)

To find the first term, substitute \(i=1\) into the expression: $$a_1 = -4(1) + 9 = 5$$

Step 3: Find the nth term, \(a_n\)

To find the nth (12th) term, substitute \(i=12\) into the expression: $$a_n = -4(12) + 9 = -39$$

Step 4: Apply the sum of arithmetic sequence formula

Using the formula for the sum of an arithmetic sequence, \(S_n = \frac{n(a_1 + a_n)}{2}\), substitute the values \(n=12\), \(a_1=5\), and \(a_n=-39\): $$S_{12} = \frac{12(5 + (-39))}{2}$$

Step 5: Calculate the sum

Now, simplify the expression: $$S_{12} = \frac{12(-34)}{2} = -204$$ So, the sum is: $$\sum_{i=1}^{12}(-4 i+9) = -204$$

Key Concepts

sum of sequence

Finding the sum of a sequence involves adding all the terms of that sequence together. For arithmetical sequences, this can be efficiently calculated using a specific technique. An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference.
In the context of our exercise, you were given the task of finding the sum of an arithmetic sequence where each term is described by the expression \(-4i + 9\). To solve it, instead of adding each term manually, a formula simplifies this process.
By determining the first term and the last term of the sequence, you can use a clever formula to find the total sum quickly.

arithmetic series formula

The arithmetic series formula is a mathematical tool that allows us to add up all the terms in an arithmetic sequence without having to do each addition separately. This formula is particularly useful because it operates by using the known values of the sequence.
The formula is: \[S_n = \frac{n(a_1 + a_n)}{2}\]
Here,

• \(S_n\) represents the sum of the sequence.
• \(n\) is the number of terms.
• \(a_1\) is the first term.
• \(a_n\) is the last term.

In the given problem, \(n = 12\), \(a_1 = 5\), and \(a_n = -39\). Plugging these into the formula allowed us to calculate the sum quickly as \(-204\). This method not only saves time but also minimizes the possibility of calculation errors.

algebraic expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation signs. In this exercise, each term of the sequence is created from an algebraic expression: \(-4i + 9\).
Understanding how to manipulate and evaluate algebraic expressions is a crucial skill, especially when dealing with sequences represented in a generalized form. Here:

• The variable \(i\) corresponds to each term's specific place in the sequence.
• \(-4\) is the coefficient that shows how each term is decreasing as you move along the sequence.
• The constant \(+9\) adjusts each term by a fixed amount.

By substituting different integer values of \(i\) from 1 to 12 into the expression, we generate the terms of the sequence. Understanding these expressions simplifies finding both individual terms and their overall sum.