In a geometric progression (GP), the relationship between the terms can be described using the common ratio \( r \).
Let the first term \( a = 7 \) and the fourth term \( a_4 = 189 \). The fourth term can be expressed in terms of the first term and the common ratio as follows:
\[ a_4 = a \cdot r^3 \]
Substituting the known values:
\[ 189 = 7 \cdot r^3 \]
To isolate \( r^3 \), divide both sides by 7:
\[ r^3 = \frac{189}{7} = 27 \]
Now, take the cube root of both sides to find \( r \):
\[ r = \sqrt[3]{27} = 3 \]
Now we have the common ratio \( r = 3 \). We can find the second and third terms of the GP:
- The second term \( a_2 \):
\[ a_2 = a \cdot r = 7 \cdot 3 = 21 \]
- The third term \( a_3 \):
\[ a_3 = a \cdot r^2 = 7 \cdot 3^2 = 7 \cdot 9 = 63 \]
Now we can calculate the sum of the first three terms:
\[ S = a + a_2 + a_3 = 7 + 21 + 63 \] \[ S = 91 \]
Thus, the sum of the first three terms of the progression is \( \boxed{91} \).
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