Problem 24 - Algebra 2

Question

For each sum, find the number of terms, the first term, and the last term. Then evaluate the series. $$ \sum_{n=5}^{10}(20-n) $$

Step-by-Step Solution

Answer

The number of terms is 6, the first term is 15, the last term is 10, and the sum of the series is 75.

Step 1: Find the Number of Terms

The number of terms can be found by subtracting the lower limit from the upper limit and then adding 1. So it's \(10 - 5 + 1 = 6\).

Step 2: Find the First Term

To find the first term, plug in the lower limit (5) to the general term formula (20 - n). This gives us \(20 - 5 = 15\). So, the first term is 15.

Step 3: Find the Last Term

Similarly, find the last term by substituting the upper limit (10) into the general term formula. This gives \(20 - 10 = 10\). So, the last term is 10.

Step 4: Calculate the Sum of the Series

Finally, we calculate the sum of the series. In an arithmetic series, the sum 'S' of 'n' terms can be calculated with the formula \(S = n/2 * (a + l)\), where 'a' is the first term, 'l' is the last term. Substituting the given values, we get \(S = 6/2 * (15 + 10) = 3 * 25 = 75\). So, the sum of the series is 75.

Key Concepts

Number of Terms

When it comes to an arithmetic series, the number of terms is an essential part of understanding the sequence, as it tells us how many elements are being summed. In the given problem, the series is represented by \( \sum_{n=5}^{10}(20-n) \). Here, it begins at 5 and ends at 10. To determine the number of terms in this series, you simply subtract the lower limit from the upper limit, which gives \( 10 - 5 \). But remember, because we're considering both endpoints, we must add 1 to this result. Therefore, the total number of terms is \( 10 - 5 + 1 = 6 \). This means the series has 6 distinct terms to consider.

First Term

Identifying the first term in an arithmetic series helps set the starting point of the sequence. From the series \( \sum_{n=5}^{10}(20-n) \), the general term formula is given as \( 20 - n \). To find the very first term, we substitute the smallest value n can take, which is the lower limit.
At \( n = 5 \), substitute into the formula:

• First term, \( a = 20 - 5 = 15 \).

Thus, the first term is 15, which begins our journey through the series.

Sum of the Series

The sum of an arithmetic series can be efficiently calculated using a straightforward formula that depends on the number of terms, the first term, and the last term. The formula is as follows:

• \( S = \frac{n}{2} \times (a + l) \)

Where \( S \) is the sum of the series, \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term. Using the values we have:

• \( n = 6 \) (the number of terms)
• \( a = 15 \) (the first term)
• \( l = 10 \) (the last term)

Substituting these into the formula gives: \[ S = \frac{6}{2} \times (15 + 10) = 3 \times 25 = 75 \]Therefore, the sum of the series, considering all the terms from first to last, is 75.

General Term Formula

The general term formula in an arithmetic series is vital as it defines the relationship between each term based on its position. In our particular sequence \( \sum_{n=5}^{10}(20-n) \), the general term formula is \( 20 - n \). This mathematical expression is the blueprint of the series, letting us calculate any term's value using its positional value of \( n \).
To understand how this works, consider each term from the 5th up to the 10th. If you want to find a specific term, such as when \( n \) is 7, you substitute

• At \( n = 7 \), term = \( 20 - 7 = 13 \).

This template allows for plug-and-play interaction for quick calculations throughout the sequence's range. It forms the backbone of analyzing terms in the complete series.