Problem 38 - College Algebra

Question

Find each indicated sum. $$\sum_{i=1}^{7} 12$$

Step-by-Step Solution

Answer

The sum \( \sum_{i=1}^{7} 12 \) equals 84.

Step 1: Understand the Sigma Notation

The exercise uses Sigma notation, which represents repeated addition. Here, the expression \( \sum_{i=1}^{7} 12 \) means to add the number '12' for each value of 'i' from 1 through 7.

Step 2: Recognize the Term to be Added

In this case, the term to be added is '12', which is a constant. That means, it does not depend on the index 'i' and is the same for each term in the sum. Therefore, we add '12' seven times.

Step 3: Calculate the Sum

With '12' added seven times, the sum is \( 12 * 7 = 84 \). Therefore, \( \sum_{i=1}^{7} 12 = 84 \).

Key Concepts

Sum of Constants

When dealing with the sum of constants in sigma notation, it’s important to realize what the notation conveys. In the expression \( \sum_{i=1}^{7} 12 \), the constant '12' is repeatedly added across a specified range for the index 'i'.
This means we are adding the number '12' a total of 7 times, once for each value from 1 to 7.

• The term '12' does not change; it remains constant regardless of 'i'.
• Consequently, the result is simply 12 times the number of terms in the sum, which in this case is 7.

In this sum, the calculation is straightforward: \( 12 \times 7 = 84 \). This concept is frequently encountered in mathematics where a constant value is summed over a range of indices.

Index Variable

The index variable, commonly denoted by 'i', is a crucial component of the sigma notation. It serves as a counter to iterate through successive terms in a series. For the given expression, \( \sum_{i=1}^{7} 12 \), 'i' takes on each integer value from 1 to 7.

• The index variable 'i' does not impact the value '12', since '12' is constant.
• However, 'i' dictates how many times the addition operation will be performed.

In our specific problem, since 'i' ranges from 1 to 7, it instructs us to add the term '12' exactly seven times. Thus, while 'i' doesn’t influence the individual terms, it is essential for understanding the limits and total count of additions in the series.

Step by Step Solution

A step-by-step solution approach is invaluable in solving mathematical expressions using sigma notation. Let’s break down the solution process: First, recognize the sigma notation \( \sum_{i=1}^{7} 12 \) represents a summation of terms for each unit step of the index variable 'i', here from 1 to 7.

• Initially, identify the constant term, which remains '12' for every incremental change in 'i'.
• Next, determine how many times this constant appears; this is equal to the number of terms dictated by the range of 'i'.
• Finally, compute the total sum: multiply the constant by the number of terms (\(12 \times 7\)).

The result of the sum is simple yet demonstrates how understanding each step can aid in solving similar problems efficiently, yielding a total of 84 for our example.