Halla La Suma De $\sum_1^{25}\left[3^{n+4}\right\].A. 1037.5 B. 1075 C. 1265.5 D. 1175

Introduction

In mathematics, a geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The sum of a geometric series can be calculated using a formula, which is a powerful tool for solving problems involving exponential growth and decay. In this article, we will explore how to calculate the sum of a geometric series, using the given problem as an example.

Understanding the Problem

The problem asks us to find the sum of the series $\sum_1^{25}\left[3^{n+4}\right]$. To solve this problem, we need to understand the concept of a geometric series and how to calculate its sum. A geometric series has the form $a + ar + ar^2 + \ldots + ar^{n-1}$, where $a$ is the first term and $r$ is the common ratio.

Breaking Down the Problem

Let's break down the given problem into smaller parts. We are given the series $\sum_1^{25}\left[3^{n+4}\right]$. To calculate the sum of this series, we need to find the first term, the common ratio, and the number of terms.

• First Term: The first term of the series is $3^{1+4} = 3^5 = 243$.
• Common Ratio: The common ratio of the series is $3$, since each term is obtained by multiplying the previous term by $3$.
• Number of Terms: The number of terms in the series is $25$, since we are summing the terms from $n=1$ to $n=25$.

Calculating the Sum

Now that we have broken down the problem, we can calculate the sum of the series using the formula for the sum of a geometric series:

$$S_n = \frac{a(r^n - 1)}{r - 1}$$

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

Plugging in the values we found earlier, we get:

$$S_{25} = \frac{243(3^{25} - 1)}{3 - 1}$$

Simplifying the expression, we get:

$$S_{25} = \frac{243(3^{25} - 1)}{2}$$

Now, we need to calculate the value of $3^{25}$. We can do this using a calculator or by using the fact that $3^5 = 243$ and $3^{10} = 59049$. Using these values, we can calculate $3^{25}$ as follows:

$$3^{25} = (3^5)^5 \cdot 3^0 = 243^5 \cdot 1 = 243 \cdot 243 \cdot 243 \cdot 243 \cdot 243 = 1,125,899,906,842,624$$

Now, we can plug this value back into the expression for $S_{25}$:

$$S_{25} = \frac{243(1,125,899,906,842,624 - 1)}{2}$$

Simplifying the expression, we get:

$$S_{25} = \frac{243(1,125,899,906,842,623)}{2}$$

$$S_{25} = 136,765,111,111,111.5$$

Conclusion

In this article, we calculated the sum of the geometric series $\sum_1^{25}\left[3^{n+4}\right]$. We broke down the problem into smaller parts, found the first term, the common ratio, and the number of terms, and then used the formula for the sum of a geometric series to calculate the sum. The final answer is $\boxed{136,765,111,111,111.5}$.

References

• Geometric Series Formula
• Sum of a Geometric Series

Frequently Asked Questions

• What is a geometric series? A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
• How do I calculate the sum of a geometric series? You can use the formula for the sum of a geometric series, which is $S_n = \frac{a(r^n - 1)}{r - 1}$, where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.
• What is the common ratio of a geometric series? The common ratio of a geometric series is the number by which each term is multiplied to get the next term.