Problem 29 - University Calculus: Early Transcendentals

Question

Evaluate the sums. $$\begin{array}{lll}\text { a. } \sum_{k=1}^{7} 3 & \text { b. } \sum_{k=1}^{500} 7 & \text { c. } \sum_{k=3}^{264} 10 \end{array}$$

Step-by-Step Solution

Answer

The sums are 21, 3500, and 2620 respectively.

Step 1: Understanding the Problem

This exercise requires evaluating three different sums where the expression for each term in the sequence is a constant. Specifically: 1. The sum of 3, repeated as many times as specified by the index range. 2. The sum of 7, repeated from 1 to 500. 3. The sum of 10, repeated from 3 to 264.

Step 1: Sum a. \( \sum_{k=1}^{7} 3 \)

The sum \( \sum_{k=1}^{7} 3 \) involves adding the constant 3 a total of 7 times. This is calculated as:\[ 3 + 3 + 3 + 3 + 3 + 3 + 3 = 7 \times 3 \]Calculate the product:\[ 7 \times 3 = 21 \]So, the sum for part a is 21.

Step 2: Sum b. \( \sum_{k=1}^{500} 7 \)

The sum \( \sum_{k=1}^{500} 7 \) consists of the constant 7 repeated 500 times. The formula used is:\[ 500 \times 7 \]Perform the multiplication:\[ 500 \times 7 = 3500 \]So, the sum for part b is 3500.

Step 3: Sum c. \( \sum_{k=3}^{264} 10 \)

The sum \( \sum_{k=3}^{264} 10 \) involves adding the constant 10 starting from k=3 to k=264. Calculate the number of times 10 is added:Since we start at k=3 and end at k=264, the number of terms is \( 264 - 3 + 1 = 262 \).Therefore, calculate:\[ 262 \times 10 = 2620 \]So, the sum for part c is 2620.

Key Concepts

Constant Sequence

A **constant sequence** in mathematics refers to a series of numbers where each term is the same value. This means there are no fluctuations or changes in the numerical value throughout the sequence. The defining characteristic of a constant sequence is that every term is identical.
In the context of summation, like in our exercise, a constant sequence significantly simplifies calculations.

• For example, in sum a, all the terms are 3, repeated 7 times.
• In sum b, the sequence consists of the number 7, repeated 500 times.
• Finally, in sum c, the number 10 is repeated from index 3 up to 264, making 262 occurrences of the constant value 10.

Knowing that all terms in a sequence are constant allows us to use straightforward multiplication to evaluate the sum, as shown in the original solutions.

Index Range

The **index range** refers to the limits or boundaries within which the summation is to be performed. It specifies from which index (starting point) to which index (ending point) the summation should include. The range is often expressed using an index variable, usually noted as "k," with lower and upper bounds.
Consider our exercise examples for clearer understanding:

• In sum a, the index range is from 1 to 7, indicating that the sum includes 7 terms.
• In sum b, the range is from 1 to 500, meaning the summation includes 500 repetitions.
• In sum c, indices span from 3 to 264, leading to 262 terms in total.

The index range is crucial as it dictates how many times the constant value is summed within the operation.

Series Evaluation

The process of **series evaluation** involves finding the total sum of a sequence of numbers. When dealing with a constant sequence, the evaluation becomes particularly simple due to the uniformity of each term. The series can be evaluated by multiplying the constant term by the number of terms in the sequence.
This operation is clearly demonstrated in the original exercise:

• For sum a, the constant 3 is multiplied by 7, giving 21.
• For sum b, the constant 7 is multiplied by 500, resulting in 3500.
• For sum c, the constant 10 is multiplied by 262, reaching a total of 2620.

The simplicity of handling constant sequence series lies in this direct multiplication approach, where the index range directly influences the multiplicative factor. Thus, knowing the number of times the sequence repeats allows for quick and efficient evaluation of the entire series.