IMU Question 5 For the series 7, 4, 1, - 2 ... find: (i) the first term and common difference (ii) the sum of the first 6 terms. (

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Certainly! Let's analyze the given arithmetic series step by step.

**Given Series:** 7, 4, 1, -2, ...

### (i) Find the First Term and Common Difference

An **arithmetic series** is a sequence of numbers in which the difference between consecutive terms is constant. This constant is known as the **common difference**.

- **First Term ($$ a_1 $$):**
  
  The first term of the series is the first number in the sequence.
  
  $$
  a_1 = 7
  $$

- **Common Difference ($$ d $$):**

To find the common difference, subtract the first term from the second term:
  
  $$
  d = 4 - 7 = -3
  $$
  
  **Verification:**
  
  Let's confirm that the difference is consistent between subsequent terms:
  
  $$
  1 - 4 = -3 \quad \text{and} \quad -2 - 1 = -3
  $$
  
  Since the difference is consistent, $$ d = -3 $$ is indeed the common difference.

**Summary:**
- **First Term ($$ a_1 $$) = 7**
- **Common Difference ($$ d $$) = -3**

### (ii) Find the Sum of the First 6 Terms

The sum of the first $$ n $$ terms of an arithmetic series can be calculated using the formula:

$$
S_n = \frac{n}{2} \times [2a_1 + (n - 1)d]
$$

Where:
- $$ S_n $$ = Sum of the first $$ n $$ terms
- $$ a_1 $$ = First term
- $$ d $$ = Common difference
- $$ n $$ = Number of terms

**Given:**
- $$ n = 6 $$
- $$ a_1 = 7 $$
- $$ d = -3 $$

**Plugging in the values:**

$$
S_6 = \frac{6}{2} \times [2 \times 7 + (6 - 1)(-3)]
$$

$$
S_6 = 3 \times [14 + (5 \times -3)]
$$

$$
S_6 = 3 \times [14 - 15]
$$

$$
S_6 = 3 \times (-1) = -3
$$

**Alternatively, you can list the first 6 terms and add them up:**

1. $$ a_1 = 7 $$
2. $$ a_2 = 4 $$
3. $$ a_3 = 1 $$
4. $$ a_4 = -2 $$
5. $$ a_5 = -5 $$
6. $$ a_6 = -8 $$

**Sum:**

$$
S_6 = 7 + 4 + 1 + (-2) + (-5) + (-8) = 7 + 4 + 1 - 2 - 5 - 8 = -3
$$

**Summary:**
- **Sum of the first 6 terms ($$ S_6 $$) = -3**

### Final Answers:

1. **First Term ($$ a_1 $$) = 7; Common Difference ($$ d $$) = -3**
2. **Sum of the first 6 terms ($$ S_6 $$) = -3**
- **First Term:** 7 - **Common Difference:** -3 - **Sum of the first 6 terms:** -3

IMU Question 5 For the series 7, 4, 1, - 2 ... find: (i) the first term and common difference (ii) the sum of the first 6 terms. (

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