Problem 52 - Algebra 2

Question

Which expression represents a series with 12 terms? F. \(\sum_{n=3}^{12} 12 n \quad\) G. \(\sum_{n=3}^{14}\left(\frac{n+4}{2}\right) \quad\) H. \(\sum_{n=9}^{21}(3 n-6) \quad\) I. \(\sum_{n=1}^{11} \frac{n}{2}\)

Step-by-Step Solution

Answer

The expression that represents a series with 12 terms is G. \( \sum_{n=3}^{14}\left(\frac{n+4}{2}\right) \)

Step 1: Check Option F

Here, the upper index is 12 and the lower index is 3. Hence, total number of terms equals to 12 - 3 + 1 = 10. So, it doesn't represent a series with 12 terms.

Step 2: Check Option G

Here, the upper index is 14 and the lower index is 3. So, the total number of terms equals to 14 - 3 + 1 = 12. We have the required condition. So, G represents a series with 12 terms.

Step 3: Check Remaining Options

You can also check the remaining options H and I using the same principle, but since we have found our answer, these steps are not necessary.

Key Concepts

summation notation

Summation notation is a compact way to represent the addition of a series of numbers. It uses the Greek letter sigma (abla) to signify that a series of terms are being added together. The format typically looks like this: \( \sum_{n=a}^{b} f(n) \), where \( a \) is the starting index, \( b \) is the ending index, and \( f(n) \) is the function applied to each term of the series.
Summation notation is particularly useful because it succinctly expresses potentially complex series, making them easier to analyze and understand. For example, in option G from the exercise, \( \sum_{n=3}^{14}\left(\frac{n+4}{2}\right) \), the notation indicates that you will apply the function \( \frac{n+4}{2} \) to each integer \( n \) starting from 3 and ending at 14 and then sum all these resulting values.
By using summation notation, mathematicians can communicate series and sequences effectively without writing out every individual term. This makes it an essential tool in mathematics.

sequence and series

In mathematics, a sequence is a set of numbers arranged in a specific order. Sequences can be either finite or infinite, depending on the number of elements. A series, on the other hand, is the sum of the terms of a sequence. When you take the terms of a sequence and add them together, you get a series.
For instance, if you have a sequence like \( 1, 2, 3, 4, 5 \), the corresponding series is \( 1 + 2 + 3 + 4 + 5 \).
In the exercise, each option represents a series constructed from a sequence of terms generated by their respective function within the summation notation. Option G, \( \sum_{n=3}^{14}\left(\frac{n+4}{2}\right) \), takes terms of the sequence generated by \( \frac{n+4}{2} \) from \( n = 3 \) to \( n = 14 \) and sums them, creating a series.
Sequences and series are fundamental in mathematics as they are used to model and solve problems across calculus, analysis, and beyond.

number of terms in a series

To determine the number of terms in a finite series, you subtract the starting index (lower limit of summation) from the ending index (upper limit of summation) and then add one. This is because you are counting inclusively from the first term through to the last.
In the given exercise, for option G, \( \sum_{n=3}^{14}\left(\frac{n+4}{2}\right) \), the number of terms can be calculated by \( 14 - 3 + 1 = 12 \). Therefore, option G indeed has 12 terms.
This method ensures that when you're given a series described by summation notation, you can easily identify how many elements it includes. Recognizing the number of terms helps verify if the series meets specific requirements, such as having exactly 12 terms in this problem.

• Subtract starting index from ending index.
• Add 1 to include both endpoints.

Once you understand this concept, interpreting and evaluating summation notation becomes straightforward.