Consider The Following Geometric Series:${ 4 - 10 + 25 - 62.5 + 156.25 - 390.625 }$Rewrite The Series Using Sigma Notation.

Introduction to Geometric Series

A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The series can be either finite or infinite, and it is an essential concept in mathematics, particularly in algebra and calculus.

Understanding the Given Series

The given series is: $4 - 10 + 25 - 62.5 + 156.25 - 390.625$. At first glance, it may seem like a random collection of numbers, but upon closer inspection, we can see that each term is obtained by multiplying the previous term by a fixed number.

Identifying the Common Ratio

To rewrite the series using sigma notation, we need to identify the common ratio. Let's examine the series more closely:

• $4$ is the first term.
• $-10$ is obtained by multiplying $4$ by $-\frac{5}{2}$.
• $25$ is obtained by multiplying $-10$ by $-\frac{5}{2}$.
• $-62.5$ is obtained by multiplying $25$ by $-\frac{5}{2}$.
• $156.25$ is obtained by multiplying $-62.5$ by $-\frac{5}{2}$.
• $-390.625$ is obtained by multiplying $156.25$ by $-\frac{5}{2}$.

We can see that the common ratio is $-\frac{5}{2}$.

Rewriting the Series using Sigma Notation

Now that we have identified the common ratio, we can rewrite the series using sigma notation. Sigma notation is a compact way of writing a series, and it is particularly useful when dealing with infinite series.

The general form of sigma notation is: $\sum_{i=1}^{n} a_i$, where $a_i$ is the $i$th term of the series, and $n$ is the number of terms.

In our case, the series can be rewritten as: $\sum_{i=1}^{6} (-\frac{5}{2})^{i-1} \cdot 4$.

Understanding the Sigma Notation

Let's break down the sigma notation:

• $\sum_{i=1}^{6}$ means that we are summing the terms from $i=1$ to $i=6$.
• $(-\frac{5}{2})^{i-1}$ is the common ratio raised to the power of $i-1$.
• $4$ is the first term of the series.

Evaluating the Series

Now that we have rewritten the series using sigma notation, we can evaluate it. To do this, we need to calculate the sum of the terms from $i=1$ to $i=6$.

Using the formula for the sum of a geometric series, we get:

$\sum_{i=1}^{6} (-\frac{5}{2})^{i-1} \cdot 4 = 4 \cdot \frac{1 - (-\frac{5}{2})^6}{1 - (-\frac{5}{2})}$

Evaluating the expression, we get:

$4 \cdot \frac{1 - (-\frac{5}{2})^6}{1 - (-\frac{5}{2})} = 4 \cdot \frac{1 - (-\frac{5}{2})^6}{1 + \frac{5}{2}}$

$= 4 \cdot \frac{1 - (-\frac{5}{2})^6}{\frac{7}{2}}$

$= 4 \cdot \frac{2}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (

Introduction to Geometric Series

A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The series can be either finite or infinite, and it is an essential concept in mathematics, particularly in algebra and calculus.

Understanding the Given Series

The given series is: $4 - 10 + 25 - 62.5 + 156.25 - 390.625$. At first glance, it may seem like a random collection of numbers, but upon closer inspection, we can see that each term is obtained by multiplying the previous term by a fixed number.

Identifying the Common Ratio

To rewrite the series using sigma notation, we need to identify the common ratio. Let's examine the series more closely:

• $4$ is the first term.
• $-10$ is obtained by multiplying $4$ by $-\frac{5}{2}$.
• $25$ is obtained by multiplying $-10$ by $-\frac{5}{2}$.
• $-62.5$ is obtained by multiplying $25$ by $-\frac{5}{2}$.
• $156.25$ is obtained by multiplying $-62.5$ by $-\frac{5}{2}$.
• $-390.625$ is obtained by multiplying $156.25$ by $-\frac{5}{2}$.

We can see that the common ratio is $-\frac{5}{2}$.

Rewriting the Series using Sigma Notation

Now that we have identified the common ratio, we can rewrite the series using sigma notation. Sigma notation is a compact way of writing a series, and it is particularly useful when dealing with infinite series.

The general form of sigma notation is: $\sum_{i=1}^{n} a_i$, where $a_i$ is the $i$th term of the series, and $n$ is the number of terms.

In our case, the series can be rewritten as: $\sum_{i=1}^{6} (-\frac{5}{2})^{i-1} \cdot 4$.

Understanding the Sigma Notation

Let's break down the sigma notation:

• $\sum_{i=1}^{6}$ means that we are summing the terms from $i=1$ to $i=6$.
• $(-\frac{5}{2})^{i-1}$ is the common ratio raised to the power of $i-1$.
• $4$ is the first term of the series.

Evaluating the Series

Now that we have rewritten the series using sigma notation, we can evaluate it. To do this, we need to calculate the sum of the terms from $i=1$ to $i=6$.

Using the formula for the sum of a geometric series, we get:

$\sum_{i=1}^{6} (-\frac{5}{2})^{i-1} \cdot 4 = 4 \cdot \frac{1 - (-\frac{5}{2})^6}{1 - (-\frac{5}{2})}$

Evaluating the expression, we get:

$4 \cdot \frac{1 - (-\frac{5}{2})^6}{1 - (-\frac{5}{2})} = 4 \cdot \frac{1 - (-\frac{5}{2})^6}{1 + \frac{5}{2}}$

$= 4 \cdot \frac{1 - (-\frac{5}{2})^6}{\frac{7}{2}}$

$= 4 \cdot \frac{2}{7} \cdot (1 - (-\frac{5}{2})^6)$

$= \frac{8}{7} \cdot (1 - (-\frac{5}{2})^6)$

Q&A Section

Q: What is a geometric series?

A: A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q: How do I identify the common ratio in a geometric series?

A: To identify the common ratio, examine the series more closely and look for a pattern. In the given series, we can see that each term is obtained by multiplying the previous term by $-\frac{5}{2}$.

Q: What is sigma notation?

A: Sigma notation is a compact way of writing a series, and it is particularly useful when dealing with infinite series. The general form of sigma notation is: $\sum_{i=1}^{n} a_i$, where $a_i$ is the $i$th term of the series, and $n$ is the number of terms.

Q: How do I evaluate a geometric series using sigma notation?

A: To evaluate a geometric series using sigma notation, we need to calculate the sum of the terms from $i=1$ to $i=n$. Using the formula for the sum of a geometric series, we get: $\sum_{i=1}^{n} a_i = a_1 \cdot \frac{1 - r^n}{1 - r}$, where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

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

A: The formula for the sum of a geometric series is: $\sum_{i=1}^{n} a_i = a_1 \cdot \frac{1 - r^n}{1 - r}$, where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Q: Can I use sigma notation to write an infinite geometric series?

A: Yes, you can use sigma notation to write an infinite geometric series. The general form of an infinite geometric series is: $\sum_{i=1}^{\infty} a_i$, where $a_i$ is the $i$th term of the series.

Q: How do I evaluate an infinite geometric series?

A: To evaluate an infinite geometric series, we need to calculate the sum of the terms from $i=1$ to $\infty$. Using the formula for the sum of an infinite geometric series, we get: $\sum_{i=1}^{\infty} a_i = \frac{a_1}{1 - r}$, where $a_1$ is the first term, and $r$ is the common ratio.

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

A: The formula for the sum of an infinite geometric series is: $\sum_{i=1}^{\infty} a_i = \frac{a_1}{1 - r}$, where $a_1$ is the first term, and $r$ is the common ratio.

Q: Can I use sigma notation to write a finite geometric series?

A: Yes, you can use sigma notation to write a finite geometric series. The general form of a finite geometric series is: $\sum_{i=1}^{n} a_i$, where $a_i$ is the $i$th term of the series, and $n$ is the number of terms.

Q: How do I evaluate a finite geometric series?

A: To evaluate a finite geometric series, we need to calculate the sum of the terms from $i=1$ to $i=n$. Using the formula for the sum of a finite geometric series, we get: $\sum_{i=1}^{n} a_i = a_1 \cdot \frac{1 - r^n}{1 - r}$, where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

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

A: The formula for the sum of a finite geometric series is: $\sum_{i=1}^{n} a_i = a_1 \cdot \frac{1 - r^n}{1 - r}$, where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.