To solve this problem, let's denote the four terms of the arithmetic sequence as $a, a + d, a + 2d, a + 3d$ and the four terms of the geometric sequence as $b, br, br^2, br^3$.

The four-term sequence we’re asked to find is formed by adding each term of the arithmetic sequence to the corresponding term of the geometric sequence:

1. First term: $a + b$
2. Second term: $a + d + br$
3. Third term: $a + 2d + br^2$
4. Fourth term: $a + 3d + br^3$

We know:

• The first term of this resulting sequence is 21.
• The second term is 39.
• The third term is 63.

We can set up the following equations based on these values:

$a + b = 21$ $a + d + br = 39$ $a + 2d + br^2 = 63$

We need to find the fourth term, $a + 3d + br^3$.

Step 1: Solve for $a$ and $b$

Let's work with the given equations:

From the first equation: $a + b = 21$

Step 2: Use the Other Equations

Substituting into the next equations will help us determine $a$, $d$, $b$, and $r$.

After finding these values, we can calculate the fourth term