Find the sum of the Following Arithmetic Sequence. Show your solutions:

1. 1, 2, 3, 4, 5

2. 2, 5, 8, 11

3. 5, 7, 9, 11, 13, 15

4. 10, 14, 18

5. -1, 1, 3, 5, 7

6. A1 = 4     a7 = 22

7. A1 = 0     a4 = 15

8. A1 = 3     a6 = - 7

9. A1 = 12   a5 = 36

10. A1 = 20   a3 = 30

To find the sum of each arithmetic sequence, we can use the formula for the sum \( S_n \) of the first \( n \) terms of an arithmetic sequence:

\[
S_n = \frac{n}{2} \times (a_1 + a_n)
\]

where:
- \( n \) is the number of terms,
- \( a_1 \) is the first term,
- \( a_n \) is the last term.

Let’s solve each problem step-by-step.

### 1. Sequence: 1, 2, 3, 4, 5
- **First term \( a_1 \)**: 1
- **Last term \( a_n \)**: 5
- **Number of terms \( n \)**: 5

\[
S_5 = \frac{5}{2} \times (1 + 5) = \frac{5}{2} \times 6 = 15
\]

**Sum**: \( S_5 = 15 \)

### 2. Sequence: 2, 5, 8, 11
- **First term \( a_1 \)**: 2
- **Last term \( a_n \)**: 11
- **Number of terms \( n \)**: 4

\[
S_4 = \frac{4}{2} \times (2 + 11) = 2 \times 13 = 26
\]

**Sum**: \( S_4 = 26 \)

### 3. Sequence: 5, 7, 9, 11, 13, 15
- **First term \( a_1 \)**: 5
- **Last term \( a_n \)**: 15
- **Number of terms \( n \)**: 6

\[
S_6 = \frac{6}{2} \times (5 + 15) = 3 \times 20 = 60
\]

**Sum**: \( S_6 = 60 \)

### 4. Sequence: 10, 14, 18
- **First term \( a_1 \)**: 10
- **Last term \( a_n \)**: 18
- **Number of terms \( n \)**: 3

\[
S_3 = \frac{3}{2} \times (10 + 18) = \frac{3}{2} \times 28 = 42
\]

**Sum**: \( S_3 = 42 \)

### 5. Sequence: -1, 1, 3, 5, 7
- **First term \( a_1 \)**: -1
- **Last term \( a_n \)**: 7
- **Number of terms \( n \)**: 5

\[
S_5 = \frac{5}{2} \times (-1 + 7) = \frac{5}{2} \times 6 = 15
\]

**Sum**: \( S_5 = 15 \)

### 6. Sequence: \( a_1 = 4 \), \( a_7 = 22 \)
- **First term \( a_1 \)**: 4
- **Last term \( a_n \)**: 22
- **Number of terms \( n \)**: 7

\[
S_7 = \frac{7}{2} \times (4 + 22) = \frac{7}{2} \times 26 = 91
\]

**Sum**: \( S_7 = 91 \)

### 7. Sequence: \( a_1 = 0 \), \( a_4 = 15 \)
- **First term \( a_1 \)**: 0
- **Last term \( a_n \)**: 15
- **Number of terms \( n \)**: 4

\[
S_4 = \frac{4}{2} \times (0 + 15) = 2 \times 15 = 30
\]

**Sum**: \( S_4 = 30 \)

### 8. Sequence: \( a_1 = 3 \), \( a_6 = -7 \)
- **First term \( a_1 \)**: 3
- **Last term \( a_n \)**: -7
- **Number of terms \( n \)**: 6

\[
S_6 = \frac{6}{2} \times (3 + (-7)) = 3 \times (-4) = -12
\]

**Sum**: \( S_6 = -12 \)

### 9. Sequence: \( a_1 = 12 \), \( a_5 = 36 \)
- **First term \( a_1 \)**: 12
- **Last term \( a_n \)**: 36
- **Number of terms \( n \)**: 5

\[
S_5 = \frac{5}{2} \times (12 + 36) = \frac{5}{2} \times 48 = 120
\]

**Sum**: \( S_5 = 120 \)

### 10. Sequence: \( a_1 = 20 \), \( a_3 = 30 \)
- **First term \( a_1 \)**: 20
- **Last term \( a_n \)**: 30
- **Number of terms \( n \)**: 3

\[
S_3 = \frac{3}{2} \times (20 + 30) = \frac{3}{2} \times 50 = 75
\]

**Sum**: \( S_3 = 75 \)

---

Would you like more details on any specific solution or have any questions?

### Related Questions:
1. How do you determine the common difference in an arithmetic sequence?
2. How can you find the nth term of an arithmetic sequence?
3. What happens if the first term of an arithmetic sequence is zero?
4. How does changing the number of terms affect the sum of the sequence?
5. Can you derive the sum formula for an arithmetic sequence?

**Tip**: Always verify that you've correctly identified the number of terms when using the sum formula for arithmetic sequences.

Learn how to find the sum of arithmetic sequences with step-by-step solutions. Includes calculations for sequences such as 1, 2, 3, 4, 5 and more. Suitable for high school students studying arithmetic sequences and series.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation