1.4 Atlehang generates a sequence, which he claims that the sequence can be arithmetic and also be geometric if and only if the first term in the sequence I. Is he correct? Motivate your answer by showing all your calculations

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Atlehang's claim is partially correct, but it requires precise conditions to hold true. Let's analyze when a sequence can be both **arithmetic** and **geometric**.

### Definitions

1. **Arithmetic Sequence**: A sequence where each term increases by a constant difference $$ d $$.
   $$
   a_n = a_1 + (n - 1)d
   $$
   
2. **Geometric Sequence**: A sequence where each term is multiplied by a constant ratio $$ r $$.
   $$
   g_n = a_1 \cdot r^{(n - 1)}
   $$

### Finding Common Conditions

For a sequence to be both arithmetic and geometric, the formulas for the $$ n $$-th term must be equal for all $$ n $$:
$$
a_1 + (n - 1)d = a_1 \cdot r^{(n - 1)}
$$

Let's analyze this equality for the first few terms.

1. **First Term ($$ n = 1 $$)**:
   $$
   a_1 = a_1 \cdot r^0 \implies a_1 = a_1
   $$
   This holds true for any $$ a_1 $$.

2. **Second Term ($$ n = 2 $$)**:
   $$
   a_1 + d = a_1 \cdot r
   $$
   Solving for $$ d $$:
   $$
   d = a_1 (r - 1)
   $$

3. **Third Term ($$ n = 3 $$)**:
   $$
   a_1 + 2d = a_1 \cdot r^2
   $$
   Substitute $$ d = a_1 (r - 1) $$:
   $$
   a_1 + 2a_1(r - 1) = a_1 r^2
   $$
   $$
   a_1 (1 + 2r - 2) = a_1 r^2
   $$
   $$
   2r - 1 = r^2
   $$
   $$
   r^2 - 2r + 1 = 0
   $$
   $$
   (r - 1)^2 = 0 \implies r = 1
   $$

### Implications of $$ r = 1 $$

Substituting $$ r = 1 $$ back into the expression for $$ d $$:
$$
d = a_1 (1 - 1) = 0
$$

This means:
- **Common Difference ($$ d $$)**: Must be **0**.
- **Common Ratio ($$ r $$)**: Must be **1**.

### Conclusion

The only way a sequence can be both arithmetic and geometric **for all terms** is if:
- The **common difference** $$ d = 0 $$, and
- The **common ratio** $$ r = 1 $$.

This results in a **constant sequence**, where every term is equal to the first term $$ a_1 $$:
$$
a_n = a_1 \quad \text{for all } n
$$

### Summary

- **Atlehang's Claim**: A sequence can be both arithmetic and geometric **if and only if** the first term satisfies certain conditions.
- **Correct Interpretation**: The sequence can only be both arithmetic and geometric **if and only if** it is a **constant sequence** (i.e., all terms are equal), which occurs when $$ d = 0 $$ and $$ r = 1 $$.

Therefore, **Atlehang is correct** in his claim **only** under the condition that the sequence is constant.
Atlehang is correct only if the sequence is constant, meaning all terms are the same.

1.4 Atlehang generates a sequence, which he claims that the sequence can be arithmetic and also be geometric if and only if the first term in the sequence I. Is he correct? Motivate your answer by showing all your calculations

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