Given:
- The sum of three numbers in a geometric progression (G.P) is 70.
- If the two extreme numbers are multiplied by 4 and the mean number is multiplied by 5, the resulting numbers are in arithmetic progression (A.P).

To Find:
The three numbers in the G.P.

Explanation:

Let's assume the three numbers in the G.P. to be a, ar, and ar^2, where 'a' is the first term and 'r' is the common ratio.

1. Sum of the three numbers:
According to the given information, the sum of the three numbers is 70. Therefore, we have the equation:
a + ar + ar^2 = 70

2. Multiplying the two extreme numbers by 4 and the mean number by 5:
We need to find a relationship between the original numbers (a, ar, ar^2) and the resulting numbers in the A.P. when multiplied by 4 and 5. Let's denote the resulting numbers as 4a, 5ar, and 4ar^2.
We know that the resulting numbers are in A.P., so we can write:
(4a) + (5ar) + (4ar^2) = a + ar + ar^2

Now, let's solve these two equations to find the values of 'a' and 'r'.

3. Solving the equations:
Simplifying the equation (4a) + (5ar) + (4ar^2) = a + ar + ar^2, we get:
4a + 5ar + 4ar^2 = a + ar + ar^2
3a + 4ar + 4ar^2 = 0

Simplifying the equation a + ar + ar^2 = 70, we get:
a(1 + r + r^2) = 70

Dividing the second equation by a, we have:
1 + r + r^2 = 70/a ----(3)

Now, substituting (3) into the first equation, we get:
3a + 4ar + 4ar^2 = 0
3a + 4ar + 4(70/a) = 0

Multiplying through by 'a', we have:
3a^2 + 4a^2r + 280 = 0

Rearranging the equation, we get:
7a^2r + 3a^2 = -280 ----(4)

Now, we have two equations: (3) and (4). By solving these equations simultaneously, we can find the values of 'a' and 'r'.

4. Solving the simultaneous equations:
By substituting (3) into (4), we get:
7a^2r + 3a^2 = -280
7(70) + 3a^2 = -280
490 + 3a^2 = -280
3a^2 = -770
a^2 = -770/3
a^2 = -256.67 (approx.)