https://file.aiquickdraw.com/mathbot/file/17260419589974sl8subr.jpg

Let's solve each problem step by step.

### Part I: Find the sum of each finite geometric sequence.
The formula to find the sum \( S_n \) of a finite geometric sequence is:

\[
S_n = \frac{a(1 - r^n)}{1 - r}
\]
where:
- \( a \) is the first term,
- \( r \) is the common ratio, and
- \( n \) is the number of terms.

#### 1. \( 1, 2, 4, 8, \dots, n = 6 \)
- First term \( a = 1 \),
- Common ratio \( r = \frac{2}{1} = 2 \),
- Number of terms \( n = 6 \).

Using the formula:

\[
S_6 = \frac{1(1 - 2^6)}{1 - 2} = \frac{1(1 - 64)}{-1} = 63
\]

#### 2. \( 2, -10, 50, -250, \dots, n = 8 \)
- First term \( a = 2 \),
- Common ratio \( r = \frac{-10}{2} = -5 \),
- Number of terms \( n = 8 \).

Using the formula:

\[
S_8 = \frac{2(1 - (-5)^8)}{1 - (-5)} = \frac{2(1 - 390625)}{6} = \frac{2(-390624)}{6} = -130208
\]

#### 3. \( 1, -4, 16, -64, \dots, n = 9 \)
- First term \( a = 1 \),
- Common ratio \( r = \frac{-4}{1} = -4 \),
- Number of terms \( n = 9 \).

Using the formula:

\[
S_9 = \frac{1(1 - (-4)^9)}{1 - (-4)} = \frac{1(1 + 262144)}{5} = \frac{262145}{5} = 52429
\]

#### 4. \( -2, -6, -18, -54, \dots, n = 9 \)
- First term \( a = -2 \),
- Common ratio \( r = \frac{-6}{-2} = 3 \),
- Number of terms \( n = 9 \).

Using the formula:

\[
S_9 = \frac{-2(1 - 3^9)}{1 - 3} = \frac{-2(1 - 19683)}{-2} = \frac{-2(-19682)}{-2} = 19682
\]

#### 5. \( 1, -5, 25, -125, \dots, n = 7 \)
- First term \( a = 1 \),
- Common ratio \( r = \frac{-5}{1} = -5 \),
- Number of terms \( n = 7 \).

Using the formula:

\[
S_7 = \frac{1(1 - (-5)^7)}{1 - (-5)} = \frac{1(1 + 78125)}{6} = \frac{78126}{6} = 13021
\]

---

### Part II: Find the sum of each infinite geometric sequence.
The sum \( S \) of an infinite geometric sequence can be found using the formula:

\[
S = \frac{a}{1 - r}
\]
where \( |r| < 1 \).

#### 1. \( 1, 0.5, 0.25, 0.125, \dots \)
- First term \( a = 1 \),
- Common ratio \( r = \frac{0.5}{1} = 0.5 \).

Using the formula:

\[
S = \frac{1}{1 - 0.5} = \frac{1}{0.5} = 2
\]

#### 2. \( 3, \frac{9}{4}, \frac{27}{16}, \frac{81}{64}, \dots \)
- First term \( a = 3 \),
- Common ratio \( r = \frac{\frac{9}{4}}{3} = \frac{3}{4} \).

Using the formula:

\[
S = \frac{3}{1 - \frac{3}{4}} = \frac{3}{\frac{1}{4}} = 12
\]

#### 3. \( 81, -27, 9, -3, \dots \)
- First term \( a = 81 \),
- Common ratio \( r = \frac{-27}{81} = -\frac{1}{3} \).

Using the formula:

\[
S = \frac{81}{1 - \left(-\frac{1}{3}\right)} = \frac{81}{\frac{4}{3}} = \frac{81 \times 3}{4} = 60.75
\]

#### 4. \( 1, -0.6, 0.36, -0.216, \dots \)
- First term \( a = 1 \),
- Common ratio \( r = \frac{-0.6}{1} = -0.6 \).

Using the formula:

\[
S = \frac{1}{1 - (-0.6)} = \frac{1}{1.6} = 0.625
\]

#### 5. \( 5, \frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \dots \)
- First term \( a = 5 \),
- Common ratio \( r = \frac{\frac{5}{2}}{5} = \frac{1}{2} \).

Using the formula:

\[
S = \frac{5}{1 - \frac{1}{2}} = \frac{5}{\frac{1}{2}} = 10
\]

---

Do you have any questions or would you like further details on any of the steps?

Here are five related questions to practice:
1. How do you determine the common ratio in a geometric sequence?
2. What is the condition for an infinite geometric series to have a sum?
3. How does the number of terms \( n \) affect the sum of a finite geometric sequence?
4. Can you derive the sum formula for a finite geometric sequence from the partial sums method?
5. How do you find the sum of a geometric sequence when the first term is negative?

**Tip:** Always check if the common ratio's absolute value is less than 1 when solving for the sum of an infinite geometric sequence.

Find the sum of each finite and infinite geometric sequence.

This math problem explores the sum of both finite and infinite geometric sequences, covering key concepts such as the geometric sequence summation formulas. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation