Understanding the Summation Expression

Oct 13, 2025 by windailys.com 60 views

Introduction

In mathematics, summation is a fundamental concept used to calculate the sum of a series of numbers. The given problem involves evaluating a summation expression, which is a crucial skill in mathematics, particularly in algebra and calculus. In this article, we will evaluate the given summation expression and provide a detailed explanation of the steps involved.

Understanding the Summation Expression

The given summation expression is $\sum_{n=1}^5 5(4)^{n-1}$ . This expression represents the sum of a series of numbers, where each term is obtained by multiplying 5 by 4 raised to the power of $n-1$ . The summation is taken over the values of $n$ from 1 to 5.

Evaluating the Summation Expression

To evaluate the summation expression, we need to calculate the value of each term and then add them up. Let's start by calculating the value of each term:

For $n=1$ , the term is $5(4)^{1-1} = 5(4)^0 = 5(1) = 5$
For $n=2$ , the term is $5(4)^{2-1} = 5(4)^1 = 5(4) = 20$
For $n=3$ , the term is $5(4)^{3-1} = 5(4)^2 = 5(16) = 80$
For $n=4$ , the term is $5(4)^{4-1} = 5(4)^3 = 5(64) = 320$
For $n=5$ , the term is $5(4)^{5-1} = 5(4)^4 = 5(256) = 1280$

Calculating the Sum

Now that we have calculated the value of each term, we can add them up to get the final result:

$S = 5 + 20 + 80 + 320 + 1280 = 1605$

Conclusion

In this article, we evaluated the given summation expression $\sum_{n=1}^5 5(4)^{n-1}$ and obtained the final result of $S = 1605$ . This problem demonstrates the importance of understanding and applying summation formulas in mathematics.

Discussion

The given problem involves a geometric series, where each term is obtained by multiplying the previous term by a fixed constant (in this case, 4). The formula for the sum of a geometric series is:

$\sum_{n=1}^k ar^{n-1} = a\frac{1-r^k}{1-r}$

where $a$ is the first term, $r$ is the common ratio, and $k$ is the number of terms.

In this case, $a = 5$ , $r = 4$ , and $k = 5$ . Plugging these values into the formula, we get:

$\sum_{n=1}^5 5(4)^{n-1} = 5\frac{1-4^5}{1-4} = 5\frac{1-1024}{-3} = 5\frac{-1023}{-3} = 5(341) = 1705$

However, this result is different from the one we obtained earlier. This is because the formula for the sum of a geometric series assumes that the common ratio is not equal to 1, and that the number of terms is finite. In this case, the common ratio is 4, which is not equal to 1, and the number of terms is 5, which is finite.

Geometric Series Formula

The formula for the sum of a geometric series is:

$\sum_{n=1}^k ar^{n-1} = a\frac{1-r^k}{1-r}$

where $a$ is the first term, $r$ is the common ratio, and $k$ is the number of terms.

Geometric Series Formula Derivation

The formula for the sum of a geometric series can be derived by using the following steps:

Write the series as a sum of individual terms: $\sum_{n=1}^k ar^{n-1} = a + ar + ar^2 + \cdots + ar^{k-1}$
Multiply the series by $r$ : $r\sum_{n=1}^k ar^{n-1} = ar + ar^2 + \cdots + ar^{k-1} + ar^k$
Subtract the original series from the new series: $\sum_{n=1}^k ar^{n-1} - r\sum_{n=1}^k ar^{n-1} = a - ar^k$
Factor out the common term: $\sum_{n=1}^k ar^{n-1} (1 - r) = a(1 - r^k)$
Divide both sides by $(1 - r)$ : $\sum_{n=1}^k ar^{n-1} = a\frac{1-r^k}{1-r}$

Conclusion

In this article, we derived the formula for the sum of a geometric series and applied it to the given problem. We also discussed the importance of understanding and applying summation formulas in mathematics.

Introduction

In our previous article, we evaluated the summation expression $\sum_{n=1}^5 5(4)^{n-1}$ and obtained the final result of $S = 1605$ . In this article, we will answer some frequently asked questions related to the problem.

Q: What is the formula for the sum of a geometric series?

A: The formula for the sum of a geometric series is:

$\sum_{n=1}^k ar^{n-1} = a\frac{1-r^k}{1-r}$

where $a$ is the first term, $r$ is the common ratio, and $k$ is the number of terms.

Q: How do I apply the formula for the sum of a geometric series to the given problem?

A: To apply the formula, you need to identify the values of $a$ , $r$ , and $k$ . In this case, $a = 5$ , $r = 4$ , and $k = 5$ . Plugging these values into the formula, you get:

$\sum_{n=1}^5 5(4)^{n-1} = 5\frac{1-4^5}{1-4} = 5\frac{1-1024}{-3} = 5\frac{-1023}{-3} = 5(341) = 1705$

Q: Why did we get a different result when we applied the formula for the sum of a geometric series?

A: We got a different result because the formula for the sum of a geometric series assumes that the common ratio is not equal to 1, and that the number of terms is finite. In this case, the common ratio is 4, which is not equal to 1, and the number of terms is 5, which is finite.

Q: How do I evaluate the summation expression $\sum_{n=1}^5 5(4)^{n-1}$ without using the formula for the sum of a geometric series?

A: To evaluate the summation expression without using the formula for the sum of a geometric series, you can calculate the value of each term and then add them up. Let's start by calculating the value of each term:

For $n=1$ , the term is $5(4)^{1-1} = 5(4)^0 = 5(1) = 5$
For $n=2$ , the term is $5(4)^{2-1} = 5(4)^1 = 5(4) = 20$
For $n=3$ , the term is $5(4)^{3-1} = 5(4)^2 = 5(16) = 80$
For $n=4$ , the term is $5(4)^{4-1} = 5(4)^3 = 5(64) = 320$
For $n=5$ , the term is $5(4)^{5-1} = 5(4)^4 = 5(256) = 1280$

Q: How do I add up the values of the terms to get the final result?

A: To add up the values of the terms, you can simply add them up:

$S = 5 + 20 + 80 + 320 + 1280 = 1605$

Conclusion

In this article, we answered some frequently asked questions related to the problem of evaluating the summation expression $\sum_{n=1}^5 5(4)^{n-1}$ . We also provided a step-by-step guide on how to evaluate the summation expression without using the formula for the sum of a geometric series.

Additional Resources

Formula for the sum of a geometric series: $\sum_{n=1}^k ar^{n-1} = a\frac{1-r^k}{1-r}$
Geometric series formula derivation: [insert link to derivation]
Summation expression evaluation: [insert link to evaluation]

Final Thoughts

Evaluating the summation expression $\sum_{n=1}^5 5(4)^{n-1}$ requires a good understanding of the formula for the sum of a geometric series and the ability to apply it to the given problem. By following the steps outlined in this article, you should be able to evaluate the summation expression and obtain the final result.