Problem 46 - Algebra and Trigonometry

Question

\(41-48\) Find the sum. $$ \sum_{i=4}^{12} 10 $$

Step-by-Step Solution

Answer

The sum is 90.

Step 1: Understanding the Summation Notation

The expression \( \sum_{i=4}^{12} 10 \) is a summation where we need to sum the number 10 from \( i = 4 \) to \( i = 12 \). Here, \( i \) is an index that starts from 4 and increments in steps of 1 until it reaches 12.

Step 2: Determine the Number of Terms

To find the number of terms in the summation, calculate the number of integers from 4 to 12, inclusive. This can be determined by subtracting 4 from 12 and adding 1, which gives us \( 12 - 4 + 1 = 9 \) terms.

Step 3: Calculate the Total Sum

Since each term in the summation is 10 and there are 9 terms, the total sum is given by multiplying 10 by the number of terms. Therefore, the total sum is \( 10 \times 9 = 90 \).

Key Concepts

Summation Notation

Summation notation is a concise way to express the sum of a sequence of numbers. It's represented by the Greek letter sigma, \( \sum \). The expression \( \sum_{i=4}^{12} 10 \) tells us two important things: what we are summing and over what range.
In this notation:

• \( i \) is called the index of summation; it indicates the variable part of the sum.
• The range of the sum is from \( i=4 \) to \( i=12 \), which are specified below and above the sigma symbol, respectively.
• The sequence being summed is the constant number 10.

Summation notation is very useful for handling large sums with multiple terms, making complex expressions much easier to manage and work with.

Index in Summation

The index in the summation notation serves as a counter or a variable that changes within a specified range. For \( \sum_{i=4}^{12} 10 \), \( i \) is the index which runs from 4 to 12. Here's how it works:

• The starting point of the index is given by the number directly below the \( \sum \) symbol, which is 4 in this case.
• The endpoint of the index is given by the number directly above the \( \sum \) symbol, which is 12 here.
• As \( i \) changes from 4 to 12, it effectively counts the terms being summed.

The index helps define each step in the summation process. It shows how values are substituted into the expression, determining the exact terms that need to be summed.

Calculating Number of Terms

To perform a summation, knowing how many terms are involved is crucial. The formula \( 12 - 4 + 1 \) is used to calculate the total number of terms in the series from \( i=4 \) to \( i=12 \). Here's why the formula works:

• The "-4" subtracts the starting index from the ending index to gauge the distance between them in terms of position.
• The "+1" ensures that both the start and end indices are included in the count, as both count as part of the sequence.
• For \( i=4 \) to \( i=12 \), this gives us \( 9 \) terms in total.

Understanding this calculation is vital since it directly influences the computation of the sum. For instance, if every term in the summation is \( 10 \), and we have calculated \( 9 \) terms, then the sum of this sequence is \( 10 \times 9 = 90 \). This illustrates how knowing the number of terms helps in determining the overall sum of the sequence.