In a geometric progression (GP), the terms can be represented as follows: let the first term be \( a \), the second term be \( ar \) (where \( r \) is the common ratio), and the third term be \( ar^2 \).
Given:
- The first term \( a = 5 \)
- The third term \( ar^2 = 245 \)
From the third term, we can substitute the value of \( a \):
\[ 5r^2 = 245 \]
To solve for \( r^2 \), we divide both sides by 5:
\[ r^2 = \frac{245}{5} = 49 \]
Now, we can find \( r \) by taking the square root:
\[ r = \sqrt{49} = 7 \quad \text{or} \quad r = -\sqrt{49} = -7 \]
Now we will calculate the middle term, \( ar \):
-
For \( r = 7 \): \[ ar = 5 \times 7 = 35 \]
-
For \( r = -7 \): \[ ar = 5 \times (-7) = -35 \]
Thus, the two possible values of the middle number are:
\[ \boxed{35} \quad \text{and} \quad \boxed{-35} \]