Problem 33 - Algebra and Trigonometry with Analytic Geometry

Question

\(\sum_{k=1}^{5}(2 k-7) \quad\)

Step-by-Step Solution

Answer

The sum is -5.

Step 1: Understand the Sigma Notation

The expression given is a Sigma (Σ) notation, which denotes a sum. The index of summation is \(k\), starting from 1 and ending at 5. The expression \( (2k - 7) \) is what we substitute \(k\) into and then sum the results.

Step 2: Substitute Values and Calculate

We substitute each integer value from 1 to 5 into the expression \( (2k - 7) \) and then sum each result. This involves calculating: \((2(1) - 7) , (2(2) - 7) , (2(3) - 7) , (2(4) - 7) , (2(5) - 7)\).

Step 3: Perform the Calculations

Perform each calculation individually:- For \( k = 1 \): \(2(1) - 7 = 2 - 7 = -5\)- For \( k = 2 \): \(2(2) - 7 = 4 - 7 = -3\)- For \( k = 3 \): \(2(3) - 7 = 6 - 7 = -1\)- For \( k = 4 \): \(2(4) - 7 = 8 - 7 = 1\)- For \( k = 5 \): \(2(5) - 7 = 10 - 7 = 3\)

Step 4: Sum the Results

Add up the results of each calculation: \(-5 + (-3) + (-1) + 1 + 3\). The sum is: \(-5 - 3 - 1 + 1 + 3 = -5\).

Key Concepts

Mathematical Summation

Mathematical summation is the process of adding a sequence of numbers. It is often represented by the Greek letter Sigma (Σ) and is used to denote the sum of terms in a sequence or series. When you see an expression in Sigma notation, like \[\sum_{k=1}^{5}(2k-7),\]it indicates that you need to replace the variable with sequential numbers and then add the results together.

In this expression, the index of summation starts at 1 and ends at 5, meaning we'll substitute these integers into the formula (2k - 7). Each substitution yields a value, and we simply add these values together to find the total sum.

Sigma notation is a concise way of expressing long sums, allowing you to efficiently handle complex algebraic expressions. It's frequently used in mathematics to simplify the representation of sequences, especially when handling arithmetic and geometric series.

Arithmetic Sequences

Arithmetic sequences are patterns of numbers with a constant difference between consecutive terms. For example, the terms created from substituting different values of \(k\) in our original problem results in the sequence -5, -3, -1, 1, 3. Notice how each term is formed by adding 2, which is the common difference.

To better understand arithmetic sequences:

• Each number in the sequence after the first is obtained by adding the common difference to the previous number.
• The expression for the nth term of an arithmetic sequence is usually given by \(a_n = a_1 + (n - 1)d\), where \(a_1\) is the first term and \(d\) is the common difference.

In our case, the formula generates terms based on the variable \(k\), which we plugged into the equation \(2k - 7\). Understanding this helps in recognizing patterns and predicting future terms without manual addition.

Index of Summation

The index of summation, often symbolized as \(k\), is an integral component of Sigma notation. It's the variable that takes on a range of integer values, guiding the summation process step by step. The index tells you the range of numbers to substitute into the expression within the Sigma, contributing to each step of the summation.

For our exercise:

• The index of summation starts at 1 and ends at 5.
• As \(k\) goes from 1 to 5, it is substituted into the sub-expression \(2k - 7\).
• Each result is computed to form a part of the overall summation.

Recognizing the role of the index of summation can greatly enhance your understanding of how and why we calculate these sums. It's particularly useful not just in arithmetic sequences but also in complex series and mathematical proofs where controlled summation is essential.