EXERCISE 5
(a) Calculate the sum of each of the following geometric series:
(1) (to 10 terms)
(2) (to 15 terms)
(3) (to 8 terms)
(4) (to 10 terms)
(5)
(to 20 terms)
(6) (to 8 terms)

Ask by Clarke Sherman. in South Africa

Feb 02,2025

Question

EXERCISE 5
(a) Calculate the sum of each of the following geometric series:
(1) (to 10 terms)
(2) (to 15 terms)
(3) (to 8 terms)
(4) (to 10 terms)
(5)
(to 20 terms)
(6) (to 8 terms)

Ask by Clarke Sherman. in South Africa

Feb 02,2025

UpStudy AI · Accepted Answer

Certainly! Let's solve each part of Exercise 5 (a) step by step.

---

### **(a) Calculate the sum of each of the following geometric series:**

#### **(1) $$ 3 + 12 + 48 + \ldots $$ (to 10 terms)**

**Identifying the Geometric Series Parameters:**
- **First term ($$ a $$)**: 3
- **Common ratio ($$ r $$)**: To find $$ r $$, divide the second term by the first term.
  
  $$
  r = \frac{12}{3} = 4
  $$
  
- **Number of terms ($$ n $$)**: 10

**Sum of a Geometric Series Formula:**

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

**Calculating the Sum:**
  
$$
S_{10} = 3 \times \frac{4^{10} - 1}{4 - 1} = 3 \times \frac{1,048,576 - 1}{3} = 3 \times \frac{1,048,575}{3} = 1,048,575
$$

**Answer:**
$$
S_{10} = 1,\!048,\!575
$$

---

#### **(2) $$ 4 - 8 + 16 - \ldots $$ (to 15 terms)**

**Identifying the Geometric Series Parameters:**
- **First term ($$ a $$)**: 4
- **Common ratio ($$ r $$)**: 
  $$
  r = \frac{-8}{4} = -2
  $$
  
- **Number of terms ($$ n $$)**: 15

**Sum of a Geometric Series Formula:**

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

**Calculating the Sum:**

$$
S_{15} = 4 \times \frac{(-2)^{15} - 1}{-2 - 1} = 4 \times \frac{-32,\!768 - 1}{-3} = 4 \times \frac{-32,\!769}{-3} = 4 \times 10,\!923 = 43,\!692
$$

**Answer:**
$$
S_{15} = 43,\!692
$$

---

#### **(3) $$ 200 + 100 + 50 + \ldots $$ (to 8 terms)**

**Identifying the Geometric Series Parameters:**
- **First term ($$ a $$)**: 200
- **Common ratio ($$ r $$)**: 
  $$
  r = \frac{100}{200} = 0.5
  $$
  
- **Number of terms ($$ n $$)**: 8

**Sum of a Geometric Series Formula:**

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

**Calculating the Sum:**

$$
S_8 = 200 \times \frac{1 - (0.5)^8}{1 - 0.5} = 200 \times \frac{1 - \frac{1}{256}}{0.5} = 200 \times \frac{\frac{255}{256}}{0.5} = 200 \times \frac{255}{128} = \frac{51,\!000}{128} \approx 398.4375
$$

**Answer:**
$$
S_8 \approx 398.44
$$

*(Depending on the desired precision, you can round accordingly.)*

---

#### **(4) $$ -27 + 9 - 3 + \ldots $$ (to 10 terms)**

**Identifying the Geometric Series Parameters:**
- **First term ($$ a $$)**: -27
- **Common ratio ($$ r $$)**: 
  $$
  r = \frac{9}{-27} = -\frac{1}{3}
  $$
  
- **Number of terms ($$ n $$)**: 10

**Sum of a Geometric Series Formula:**

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

**Calculating the Sum:**

$$
S_{10} = -27 \times \frac{1 - \left(-\frac{1}{3}\right)^{10}}{1 - \left(-\frac{1}{3}\right)} = -27 \times \frac{1 - \frac{1}{59,\!049}}{1 + \frac{1}{3}} = -27 \times \frac{\frac{59,\!048}{59,\!049}}{\frac{4}{3}} = -27 \times \frac{59,\!048}{59,\!049} \times \frac{3}{4}
$$

Simplifying:

$$
S_{10} = -27 \times \frac{59,\!048}{59,\!049} \times \frac{3}{4} = -\frac{81}{4} \times \frac{59,\!048}{59,\!049} \approx -20.25 \times 0.999983 = -20.2496
$$

**Answer:**
$$
S_{10} \approx -20.25
$$

*(Again, rounding based on desired precision.)*

---

#### **(5) $$ x + x^2 + x^3 + \ldots $$ (to 20 terms)**

**Identifying the Geometric Series Parameters:**
- **First term ($$ a $$)**: $$ x $$
- **Common ratio ($$ r $$)**: $$ x $$
  
- **Number of terms ($$ n $$)**: 20

**Sum of a Geometric Series Formula:**

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

**Calculating the Sum:**

$$
S_{20} = x \times \frac{x^{20} - 1}{x - 1} = \frac{x (x^{20} - 1)}{x - 1} = \frac{x^{21} - x}{x - 1}
$$

**Answer:**
$$
S_{20} = \frac{x^{21} - x}{x - 1}
$$

*(This is the general form of the sum. If $$ |x| < 1 $$, one might consider the infinite sum, but since it's specified to 20 terms, this is the precise expression.)*

---

#### **(6) $$ x y^{10} + x y^{9} + x y^{8} + \ldots $$ (to 8 terms)**

**Identifying the Geometric Series Parameters:**
- **First term ($$ a $$)**: $$ x y^{10} $$
- **Common ratio ($$ r $$)**: To find $$ r $$, divide the second term by the first term.
  
  $$
  r = \frac{x y^{9}}{x y^{10}} = \frac{1}{y}
  $$
  
- **Number of terms ($$ n $$)**: 8

**Sum of a Geometric Series Formula:**

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

**Calculating the Sum:**

$$
S_8 = x y^{10} \times \frac{1 - \left(\frac{1}{y}\right)^8}{1 - \frac{1}{y}} = x y^{10} \times \frac{1 - y^{-8}}{1 - y^{-1}} = x y^{10} \times \frac{y^{8} - 1}{y^{8}} \times \frac{y}{y - 1}
$$

Simplifying:

$$
S_8 = x y^{10} \times \frac{y^{8} - 1}{y^{8}} \times \frac{y}{y - 1} = x y^{10} \times \frac{y^{8} - 1}{y^{7}(y - 1)} } = x \times \frac{y^{10}(y^{8} - 1)}{y^{7}(y - 1)} } = x \times \frac{y^{18} - y^{10}}{y^{7} (y - 1)} } = x \times \frac{y^{11} - y^{3}}{y - 1}
$$

Alternatively, a simpler approach:

$$
S_8 = x y^{10} \times \frac{1 - y^{-8}}{1 - y^{-1}} = x y^{10} \times \frac{y^{8} - 1}{y^{8}(1 - \frac{1}{y})} = x y^{10} \times \frac{y^{8} - 1}{y^{8}} \times \frac{y}{y - 1} = x y^{10} \times \frac{y^{8} - 1}{y^{7}(y - 1)}}
$$

Simplify:

$$
S_8 = x \frac{y^{10} (y^{8} - 1)}{y^{7} (y - 1)} } = x \frac{y^{18} - y^{10}}{y^{7} (y - 1)} } = x \frac{y^{11} - y^{3}}{y - 1}
$$

**Answer:**
$$
S_8 = \frac{x y^{11} - x y^{3}}{y - 1} = \frac{x(y^{11} - y^{3})}{y - 1}
$$

*(This is the simplified form of the sum. Depending on the context, further factoring might be possible.)*

---

### **Summary of Results:**

1. **Sum of 10 terms** $$ 3 + 12 + 48 + \ldots $$:

$$
   S_{10} = 1,\!048,\!575
   $$

2. **Sum of 15 terms** $$ 4 - 8 + 16 - \ldots $$:

$$
   S_{15} = 43,\!692
   $$

3. **Sum of 8 terms** $$ 200 + 100 + 50 + \ldots $$:

$$
   S_8 \approx 398.44
   $$

4. **Sum of 10 terms** $$ -27 + 9 - 3 + \ldots $$:

$$
   S_{10} \approx -20.25
   $$

5. **Sum of 20 terms** $$ x + x^2 + x^3 + \ldots $$:

$$
   S_{20} = \frac{x^{21} - x}{x - 1}
   $$

6. **Sum of 8 terms** $$ x y^{10} + x y^{9} + x y^{8} + \ldots $$:

$$
   S_8 = \frac{x(y^{11} - y^{3})}{y - 1}
   $$

---

If you need further clarification on any of these parts or have additional questions, feel free to ask!
Here are the sums of the geometric series: 1. $$ 3 + 12 + 48 + \ldots $$ (10 terms): 1,048,575 2. $$ 4 - 8 + 16 - \ldots $$ (15 terms): 43,692 3. $$ 200 + 100 + 50 + \ldots $$ (8 terms): ≈398.44 4. $$ -27 + 9 - 3 + \ldots $$ (10 terms): ≈-20.25 5. $$ x + x^2 + x^3 + \ldots $$ (20 terms): $$ \frac{x^{21} - x}{x - 1} $$ 6. $$ x y^{10} + x y^{9} + x y^{8} + \ldots $$ (8 terms): $$ \frac{x(y^{11} - y^{3})}{y - 1} $$

EXERCISE 5
(a) Calculate the sum of each of the following geometric series:
(1) (to 10 terms)
(2) (to 15 terms)
(3) (to 8 terms)
(4) (to 10 terms)
(5)
(to 20 terms)
(6) (to 8 terms)

Upstudy AI Solution

Answer

Solution

(a) Calculate the sum of each of the following geometric series:

(1) (to 10 terms)

(2) (to 15 terms)

(3) (to 8 terms)

(4) (to 10 terms)

(5) (to 20 terms)

(6) (to 8 terms)

Summary of Results:

Beyond the Answer

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EXERCISE 5 (a) Calculate the sum of each of the following geometric series: (1) (to 10 terms) (2) (to 15 terms) (3) (to 8 terms) (4) (to 10 terms) (5) (to 20 terms) (6) (to 8 terms)

Upstudy AI Solution

Answer

Solution

(a) Calculate the sum of each of the following geometric series:

(1) (to 10 terms)

(2) (to 15 terms)

(3) (to 8 terms)

(4) (to 10 terms)

(5) (to 20 terms)

(6) (to 8 terms)

Summary of Results:

Beyond the Answer

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EXERCISE 5
(a) Calculate the sum of each of the following geometric series:
(1) (to 10 terms)
(2) (to 15 terms)
(3) (to 8 terms)
(4) (to 10 terms)
(5)
(to 20 terms)
(6) (to 8 terms)