Given the geometric series $$ a+a r+a r^{2}+a r^{3}+\ldots $$, where $$ a $$ is the first tem and $$ r $$ is the common ratio. Prove thar the sum to $$ n $$ terms of this series is given by $$ S_{n}=\frac{a\left(r^{n}-1\right)}{r-1}: r \neq 1 $$ The first two termbs of a geometric sequence with constant ratio $$ r $$, and an arithmetic sequence with constant first difference, $$ d $$, is tise same. The first term is 12. 4.2.1 Write down the second and third terns of EACH sequence in terms of $$ d $$ and $$ r $$.

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Certainly! Let's analyze both the geometric and arithmetic sequences based on the given information.

### Given:
- **First term ($$ a $$)**: 12
- **Geometric sequence common ratio ($$ r $$)**
- **Arithmetic sequence common difference ($$ d $$)**
- **First two terms of both sequences are the same**

### Geometric Sequence:
In a geometric sequence, each term is obtained by multiplying the previous term by the common ratio $$ r $$.

1. **First Term ($$ G_1 $$)**:
   $$
   G_1 = a = 12
   $$
2. **Second Term ($$ G_2 $$)**:
   $$
   G_2 = a \cdot r = 12r
   $$
3. **Third Term ($$ G_3 $$)**:
   $$
   G_3 = G_2 \cdot r = 12r \cdot r = 12r^2
   $$

### Arithmetic Sequence:
In an arithmetic sequence, each term is obtained by adding the common difference $$ d $$ to the previous term.

1. **First Term ($$ A_1 $$)**:
   $$
   A_1 = a = 12
   $$
2. **Second Term ($$ A_2 $$)**:
   $$
   A_2 = A_1 + d = 12 + d
   $$
3. **Third Term ($$ A_3 $$)**:
   $$
   A_3 = A_2 + d = (12 + d) + d = 12 + 2d
   $$

### Summary:
- **Geometric Sequence Terms**:
  $$
  G_1 = 12,\quad G_2 = 12r,\quad G_3 = 12r^2
  $$
- **Arithmetic Sequence Terms**:
  $$
  A_1 = 12,\quad A_2 = 12 + d,\quad A_3 = 12 + 2d
  $$

### Additional Note:
Since the first two terms of both sequences are the same:
$$
G_2 = A_2 \implies 12r = 12 + d \implies d = 12(r - 1)
$$
This relationship allows you to express $$ d $$ in terms of $$ r $$ or vice versa if needed for further problems.

Answer:
After the first term of 12, the geometric sequence has second term 12·r and third term 12·r². The arithmetic sequence has second term 12 + d and third term 12 + 2d. Thus,

• Geometric terms:
 Second term = 12 r
 Third term = 12 r²

• Arithmetic terms:
 Second term = 12 + d
 Third term = 12 + 2d
The second term of the geometric sequence is $$12r$$ and the third term is $$12r^2$$. The second term of the arithmetic sequence is $$12 + d$$ and the third term is $$12 + 2d$$.

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