Question

Compute the altemating sum of all elements in a list. For example, if your program reads the input then it computes $$ \begin{array}{ccccccccc} 1 & 4 & 9 & 16 & 9 & 7 & 4 & 9 & 11 \\ 1-4 & +9 & -16 & +9 & -7+4 & -9 & +11=-2 \end{array} $$

Short answer

The alternating sum of the list is -2.

Step-by-step solution

Organize the List

Let's write down the given list of numbers: \(1, 4, 9, 16, 9, 7, 4, 9, 11\). Each element has a specific position, which will be utilized to determine whether to add or subtract the number.

Identify the Pattern

Notice the pattern: for the alternating sum, you add when the position is odd and subtract when it's even. The position is determined by the value's order in the list.

Apply the Alternating Sums

We start with the first element at position 1 (odd), adding it: \(1\). Next, subtract the second element (even): \(1 - 4 = -3\). Then add the third element: \(-3 + 9 = 6\), and continue this process.

Complete the Calculations

Continue the calculations with the pattern identified: 1. Start: \(1\)2. Subtract: \(1 - 4 = -3\)3. Add: \(-3 + 9 = 6\)4. Subtract: \(6 - 16 = -10\)5. Add: \(-10 + 9 = -1\)6. Subtract: \(-1 - 7 = -8\)7. Add: \(-8 + 4 = -4\)8. Subtract: \(-4 - 9 = -13\)9. Add: \(-13 + 11 = -2\)

Final Answer

Following the alternating sum method, the final result after including all elements is -2.

Key concepts

List Operations

To begin with calculating the alternating sum of a list, understand that list operations involve working with a collection of elements. Lists are fundamental structures in programming and mathematics. They hold a series of elements, such as numbers. You can perform several operations on these lists, such as adding elements, removing them, or manipulating them in different ways. In this task, we are particularly interested in accessing list elements based on their position.

Every element in a list has an index, which represents its position. Indexing typically starts at 0 in programming languages like Python, but in mathematical problems, we often start with 1. Understanding how to access and use these indices is crucial for performing any operations, including calculating something like an alternating sum.

Sequence Patterns

Recognizing sequence patterns is imperative in simplifying complex problems. An alternating sum follows a specific sequence pattern where you alternately add and subtract elements based on their position.

For the problem at hand, the pattern is simple but important:

• Add elements that are in odd positions.
• Subtract elements found in even positions.

This pattern continues through the entire list. Recognizing this repeating sequence allows streamlining of the calculation and application of operations. By following the alternating pattern, students can vastly reduce errors in the sum calculation process.

Index-Based Calculation

Index-based calculation is the cornerstone of solving alternating sum problems. By leveraging the index of each element within a list, you determine whether to add or subtract it. The index tells you the element's position, helping to apply the alternating pattern efficiently.

Consider the list: 1. The first element (index 1) is added. 2. The second element (index 2) is subtracted. 3. The third element (index 3) is added. 4. And so forth.

This systematic approach ensures consistency and accuracy in calculations. Transcending simply following steps, index-based calculations provide a logical and repeatable mechanism to achieve results. Relying on indices empowers you to navigate through any list, applying the set alternating operations with confidence.