To solve this problem, let's break it down step-by-step:
Step 1: Find the first term and common difference of the arithmetic sequence.
Since the common difference is 2, we can say that the arithmetic sequence is defined by the formula a_n = a_1 + (n - 1)d, where a_n represents the nth term, a_1 is the first term, n is the term number, and d is the common difference.
We know that d = 2, and it is also mentioned that the first term of the arithmetic sequence is also the first term of a geometric sequence. Let's denote this first term as 'x'.
So, a_1 = x.
To find the value of x, we need more information. Let's move to step 2.
Step 2: Find the sum of the first 3 terms of the arithmetic sequence.
The sum of the first three terms of an arithmetic sequence can be calculated using the formula S = (n/2)(2a_1 + (n - 1)d), where S represents the sum, n is the number of terms, a_1 is the first term, and d is the common difference.
In this case, n = 3 and d = 2.
Using the given information, we have:
S = (3/2)(2x + (3 - 1)2)
Simplifying this equation:
S = (3/2)(2x + 4)
S = 3x + 6
Step 3: Find the sum of the first 9 terms of the arithmetic sequence.
Using the formula for the sum of an arithmetic sequence, we can calculate the sum of the first nine terms.
In this case, n = 9. Applying the formula:
S = (9/2)(2x + (9 - 1)2)
Simplifying this equation:
S = (9/2)(2x + 16)
S = 9x + 72
Step 4: Find the second and third terms of the geometric sequence.
Now, we know that the sum of the first three terms of the arithmetic sequence, S = 3x + 6, is the second term of the geometric sequence.
Similarly, the sum of the first nine terms of the arithmetic sequence, S = 9x + 72, is the third term of the geometric sequence.
So, the second term is 3x + 6, and the third term is 9x + 72.
Therefore, the first three terms of the geometric sequence are x, 3x + 6, 9x + 72.