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An arithmetic sequence is a sequence of numbers where each term after the first is formed by adding a constant difference to the previous term. This constant is known as the 'common difference'. For example, in the sequence 3, 6, 9, 12, the common difference is 3. This means each term increases by 3 from the one before it.
In our original exercise, the sequence is given by the formula \(-2k\). When you look at the generated terms \(-2, -4, -6, ..., -14\), you can see that each term decreases by 2, demonstrating an arithmetic sequence with a common difference of -2.
Understanding arithmetic sequences helps in predicting future terms or summing a specified number of terms efficiently. Recognizing this pattern within a problem is crucial for simplifying calculations and solving problems more accurately.

Summation notation is a powerful shorthand for expressing the sum of terms in a sequence. It is denoted by the Greek capital letter Sigma \(\Sigma\). For example, the expression \( \sum_{k=1}^{7} (-2k) \) instructs you to sum the sequence \(-2k\) for all values of \(k\) from 1 to 7.
The numbers below and above the Sigma tell you where to start and stop the summation. The expression \(-2k\), the formula inside the summation, determines each term of the sequence:

• The lower limit \(k=1\) corresponds to the initial value.
• The upper limit \(k=7\) tells us when to stop calculating new terms.

This approach allows you to quickly set up and solve sums without manually writing all individual terms. Mastery of this notation opens the door to efficiently handling complex sequences and series in mathematics, making it a critical concept for any aspiring mathematician.

Evaluating sums means calculating the total for a sequence of numbers defined in summation notation. To evaluate the sum \( \sum_{k=1}^{7} (-2k) \), follow these simple steps:

• First, generate each term in the sequence by substituting values from 1 to 7 into the expression \(-2k\).
• Calculate each result: \(-2(1)=-2\), \(-2(2)=-4\), up to \(-2(7)=-14\).
• Add these values together: \((-2) + (-4) + (-6) + (-8) + (-10) + (-12) + (-14)\).
• The final answer is the total sum of these numbers.

This methodical approach ensures each term is correctly evaluated and added. It's important to follow each step carefully, ensuring precision and accuracy, which significantly aids in understanding and solving problems involving sequences and series.