Question

Three number form a GP . if the first term and the third term are 5 and 245 respectively find the two possible values of the middle number

Answer 1

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, \( ar \):

1. For \( r = 7 \):
\[
ar = 5 \times 7 = 35
\]

2. For \( r = -7 \):
\[
ar = 5 \times (-7) = -35
\]

Thus, the two possible values of the middle number are:

\[
\boxed{35} \quad \text{and} \quad \boxed{-35}
\]