Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that \(\sum_{i=1}^{10} a_{i}=40\) and \(\sum_{i=1}^{10} b_{i}=50\). Calculate each of the following. $$ \sum_{i=1}^{10}\left(a_{i}+b_{i}\right) $$

Step-by-Step Solution

Answer

The sum is 90.

Step 1: Understanding the Problem

We are given that the sum of the sequence \(a_i\) from \(i=1\) to 10 is 40, and the sum of the sequence \(b_i\) from \(i=1\) to 10 is 50. We need to find the sum of the sequence \(a_i + b_i\) from \(i=1\) to 10.

Step 2: Applying the Sum Property

We can use the property of summation: \( \sum (a_i + b_i) = \sum a_i + \sum b_i \). This means we can add the sum of the \(a_i\) sequence and the sum of the \(b_i\) sequence to find the sum of the combined sequence.

Step 3: Performing the Addition

Add the sums of the two individual sequences using the property from the previous step: \( \sum_{i=1}^{10} a_i + \sum_{i=1}^{10} b_i = 40 + 50 = 90 \).

Step 4: Interpreting the Result

The sum of \(a_i\) and \(b_i\) from \(i=1\) to 10 is equal to 90, which represents the sum of adding each corresponding element from these two sequences.

Key Concepts

Exploring Series and Sequences in Calculus

In calculus, a series is the sum of the terms of a sequence. A sequence is essentially a set of numbers arranged in a specific order. While sequences focus on the arrangement, series deal with the accumulation of values. This can be visualized as arranging dominoes (sequence), then toppling them one-by-one and seeing the total movement they create (series). Series and sequences are fundamental in understanding mathematical concepts because they provide a structured way of dissecting complex sequences of numbers.

• A sequence is an ordered list of numbers like 1, 2, 3, 4, 5.
• A series is the sum of elements of a sequence, for example, 1 + 2 + 3 + 4 + 5.

To solve problems involving deterministic patterns in mathematics, understanding how sequences assemble and how we sum them up into series is crucial. By cracking these methods, one can simplify and solve intricate mathematical problems with efficiency.

Understanding the Sum Property in Calculus

The sum property in calculus simplifies the summation of a sequence of sums. Essentially, it allows us to break down complex problems into easier parts. Simply put, when summing two separate sequences, one can sum each sequence individually and then combine the results. For example, if we want to find the total of two separate sums, we can add those sums directly.This property becomes particularly useful when evaluating the sum of mixed sequences, like in the problem \( \sum_{i=1}^{10}(a_{i} + b_{i}) \) because:

• It breaks down the sequences into manageable parts, reducing computation errors.
• It simplifies the understanding of big series by deconstructing them.

By applying the sum property, students can tackle complicated summation problems with clarity, ensuring they follow the correct methodological steps to achieve accurate results.

Educational Methods for Mastering Summation Concepts

Effective educational methods are crucial for understanding summation in calculus. Here are some approaches to enhance learning:

• Visual Aids: Creating visual representations of series and sequences helps in grasping the concepts faster. Diagrams make theoretical ideas seem more tangible.
• Practical Exercises: Regularly practicing problems, like calculating sums of sequences, sharpens skills and improves problem-solving techniques.
• Step-by-step Guidance: Breaking down complex problems into smaller steps, as demonstrated in the solution, aids in deeper conceptual understanding.

These methods not only improve comprehension but also help retain concepts longer by embedding them within practical scenarios. Utilizing a variety of educational strategies caters to different learning styles, propelling students from basic understanding to mastery of summation in calculus.