In this problem, we are working with both an arithmetic sequence and a geometric sequence. Let's break it down step by step.

Given:

• The first term of the arithmetic sequence $a_1 = 4$.
• The first, third, and seventh terms of the arithmetic sequence form consecutive terms of a geometric sequence.

Let the terms of the arithmetic sequence be denoted as: $a_n = a_1 + (n-1)d$ where $a_1 = 4$ is the first term and $d$ is the common difference.

Step 1: Express the relevant terms of the arithmetic sequence

• The first term $a_1 = 4$.
• The third term $a_3 = a_1 + 2d = 4 + 2d$.
• The seventh term $a_7 = a_1 + 6d = 4 + 6d$.

Step 2: Use the property of the geometric sequence

Since $a_1$, $a_3$, and $a_7$ form consecutive terms of a geometric sequence, we know that the ratio between consecutive terms of a geometric sequence is constant. Therefore, the following relationship holds: $\frac{a_3}{a_1} = \frac{a_7}{a_3}$

Step 3: Set up the equation

Using the expressions for $a_1$, $a_3$, and $a_7$: $\frac{4 + 2d}{4} = \frac{4 + 6d}{4 + 2d}$ Cross-multiply to get rid of the fractions: $(4 + 2d)^2 = 4(4 + 6d)$ Expand both sides: $(16 + 8d + 4d^2) = 16 + 24d$ Simplify: $4d^2 + 8d = 24d$ $4d^2 - 16d = 0$ Factor the equation: $4d(d - 4) = 0$ So, $d = 0$ or $d = 4$.

Step 4: Analyze the solutions

• If $d = 0$, then the sequence is constant, which contradicts the condition that the terms form a geometric sequence. So, we discard this solution.
• Therefore, $d = 4$.

Step 5: Find the first term and common ratio of the geometric sequence

Now, the terms of the arithmetic sequence are: $a_1 = 4, \quad a_3 = 4 + 2(4) = 12, \quad a_7 = 4 + 6(4) = 28$ These form the geometric sequence $4, 12, 28$.

Let the first term of the geometric sequence be $G_1 = 4$ and the common ratio be $r$. Since the second term is $a_3 = 12$, we can solve for $r$: $r = \frac{12}{4} = 3$

Final Answer:

a) For the arithmetic sequence:

• $a_1 = 4$
• $d = 4$

b) For the geometric sequence:

• $G_1 = 4$
• $r = 3$

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Would you like more details or have any questions?

Here are some related questions to consider:

1. How would the solution change if the first term of the arithmetic sequence were different?
2. What if the third and fifth terms of the arithmetic sequence formed a geometric sequence instead?
3. Can the common ratio $r$ of a geometric sequence ever be negative?
4. How can we generalize the method for any arithmetic sequence where some terms form a geometric sequence?
5. What would happen if more than three terms of an arithmetic sequence were in geometric progression?

Tip: In arithmetic sequences, the relationship between terms is linear, while in geometric sequences, it is exponential. This difference can help in recognizing patterns in different problems.