Explanation:

Let's solve step-by-step:

- We subtract x from each:

Numbers become: 20-x, 37-x, 54-x, 105-x.

- They are in proportion:

(20-x)/(37-x) = (54-x)/(105-x)

- Cross-multiplied:

(20-x)(105-x) = (37-x)(54-x)

- Expanding:

2100 - 20x - 105x + x² = 1998 - 37x - 54x + x²

2100 - 125x + x² = 1998 - 91x + x²

- x² cancels:

2100 - 125x = 1998 - 91x

2100 - 1998 = 125x - 91x

102 = 34x

x = 3

Now, mean proportional between (7x-5) and (x+1):

Values are 7×3-5 = 16, and 3+1 = 4

Mean proportional = v(16×4) = v64 = 8

Options:

- Option 1: 8

- Option 2: 6

- Option 3: 12

- Option 4: 9

Option 1 (8) is correct.