$$ \mathrm{T}_{n}=a+(n-1) d=12+(n-1)(5)=5 n+7 $$ EXERCISE 5 Determine the sum of the series: $$ 1.1 \quad 11+8+5+2+\ldots-166 $$. $$ 1.21+5+9+13+\ldots+113 $$ $$ 1.3-12-5+2+9+\ldots+121 $$ Determine the sum of the first 40 terms of the arithmetic series $$ 2.1 \quad 2+10+18+\ldots \quad n=40 $$ $$ 2.216+10+4-2-\ldots $$

Question

UpStudy AI · Accepted Answer

Certainly! Let's solve each part of **EXERCISE 5** step by step. We'll be dealing with arithmetic series, where each term increases or decreases by a constant difference. The general formula for the $$ n $$-th term of an arithmetic sequence is:

$$
\mathrm{T}_{n} = a + (n-1)d
$$

where:
- $$ a $$ is the first term,
- $$ d $$ is the common difference,
- $$ n $$ is the term number.

The sum of the first $$ n $$ terms of an arithmetic series is given by:

$$
S_n = \frac{n}{2} \times (a + l)
$$

where:
- $$ S_n $$ is the sum of the first $$ n $$ terms,
- $$ l $$ is the last term.

Let's proceed with each part of the exercise.

---

### **1. Determine the Sum of the Series**

#### **1.1 $$ 11 + 8 + 5 + 2 + \ldots -166 $$**

- **First Term ($$ a $$):** 11
- **Common Difference ($$ d $$):** $$ 8 - 11 = -3 $$
- **Last Term ($$ l $$):** -166

**Step 1: Find the Number of Terms ($$ n $$)**

Using the $$ n $$-th term formula:

$$
l = a + (n-1)d
$$
$$
-166 = 11 + (n-1)(-3)
$$
$$
-166 - 11 = -3(n - 1)
$$
$$
-177 = -3(n - 1)
$$
$$
n - 1 = \frac{177}{3} = 59
$$
$$
n = 60
$$

**Step 2: Calculate the Sum ($$ S_n $$)**

$$
S_{60} = \frac{60}{2} \times (11 + (-166)) = 30 \times (-155) = -4650
$$

**Answer:** The sum of the series is **-4650**.

---

#### **1.2 $$ 1 + 5 + 9 + 13 + \ldots + 113 $$**

- **First Term ($$ a $$):** 1
- **Common Difference ($$ d $$):** $$ 5 - 1 = 4 $$
- **Last Term ($$ l $$):** 113

**Step 1: Find the Number of Terms ($$ n $$)**

$$
113 = 1 + (n-1)4
$$
$$
113 - 1 = 4(n - 1)
$$
$$
112 = 4n - 4
$$
$$
116 = 4n
$$
$$
n = 29
$$

**Step 2: Calculate the Sum ($$ S_n $$)**

$$
S_{29} = \frac{29}{2} \times (1 + 113) = 14.5 \times 114 = 1653
$$

**Answer:** The sum of the series is **1653**.

---

#### **1.3 $$ -12 -5 + 2 + 9 + \ldots + 121 $$**

- **First Term ($$ a $$):** -12
- **Common Difference ($$ d $$):** $$ -5 - (-12) = 7 $$
- **Last Term ($$ l $$):** 121

**Step 1: Find the Number of Terms ($$ n $$)**

$$
121 = -12 + (n-1)7
$$
$$
121 + 12 = 7(n - 1)
$$
$$
133 = 7(n - 1)
$$
$$
n - 1 = \frac{133}{7} = 19
$$
$$
n = 20
$$

**Step 2: Calculate the Sum ($$ S_n $$)**

$$
S_{20} = \frac{20}{2} \times (-12 + 121) = 10 \times 109 = 1090
$$

**Answer:** The sum of the series is **1090**.

---

### **2. Determine the Sum of the First 40 Terms of the Arithmetic Series**

#### **2.1 $$ 2 + 10 + 18 + \ldots $$ (First 40 Terms)**

- **First Term ($$ a $$):** 2
- **Common Difference ($$ d $$):** $$ 10 - 2 = 8 $$
- **Number of Terms ($$ n $$):** 40

**Step 1: Find the 40th Term ($$ l $$)**

$$
l = a + (n-1)d = 2 + (40-1)8 = 2 + 312 = 314
$$

**Step 2: Calculate the Sum ($$ S_n $$)**

$$
S_{40} = \frac{40}{2} \times (2 + 314) = 20 \times 316 = 6320
$$

**Answer:** The sum of the first 40 terms is **6320**.

---

#### **2.2 $$ 16 + 10 + 4 - 2 - \ldots $$ (Assuming First 40 Terms)**

- **First Term ($$ a $$):** 16
- **Common Difference ($$ d $$):** $$ 10 - 16 = -6 $$
- **Number of Terms ($$ n $$):** 40

**Step 1: Find the 40th Term ($$ l $$)**

$$
l = a + (n-1)d = 16 + (40-1)(-6) = 16 - 234 = -218
$$

**Step 2: Calculate the Sum ($$ S_n $$)**

$$
S_{40} = \frac{40}{2} \times (16 + (-218)) = 20 \times (-202) = -4040
$$

**Alternatively, using the formula:**

$$
S_n = \frac{n}{2} [2a + (n-1)d]
$$
$$
S_{40} = \frac{40}{2} [2(16) + 39(-6)] = 20 [32 - 234] = 20 \times (-202) = -4040
$$

**Answer:** The sum of the first 40 terms is **-4040**.

---

### **Summary of Answers**

1. **1.1:** The sum is **-4650**.
2. **1.2:** The sum is **1653**.
3. **1.3:** The sum is **1090**.
4. **2.1:** The sum of the first 40 terms is **6320**.
5. **2.2:** The sum of the first 40 terms is **-4040**.

If you have any further questions or need clarification on any step, feel free to ask!
**Summary of Solutions:** 1. **1.1:** Sum = -4650 2. **1.2:** Sum = 1653 3. **1.3:** Sum = 1090 4. **2.1:** Sum of first 40 terms = 6320 5. **2.2:** Sum of first 40 terms = -4040

EXERCISE 5
Determine the sum of the series:
.

Determine the sum of the first 40 terms of the arithmetic series

Solución de inteligencia artificial de Upstudy

Responder

Solución

1. Determine the Sum of the Series

1.1

1.2

1.3

2. Determine the Sum of the First 40 Terms of the Arithmetic Series

2.1 (First 40 Terms)

2.2 (Assuming First 40 Terms)

Summary of Answers

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EXERCISE 5 Determine the sum of the series: . Determine the sum of the first 40 terms of the arithmetic series

Solución de inteligencia artificial de Upstudy

Responder

Solución

1. Determine the Sum of the Series

1.1

1.2

1.3

2. Determine the Sum of the First 40 Terms of the Arithmetic Series

2.1 (First 40 Terms)

2.2 (Assuming First 40 Terms)

Summary of Answers

The Deep Dive

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EXERCISE 5
Determine the sum of the series:
.

Determine the sum of the first 40 terms of the arithmetic series