Three numbers, a, b, and c are said to be in continued proportion if a : b : : b : c. What number must be added to each of the numbers 1, 7 and 25 so that the resulting sums, in that order, maybe in continued proportion?

Three numbers, a, b, and c are said to be in continued proportion if a : b : : b : c.

$$\frac{a}{b}\ =\ \frac{b}{c}$$

$$b^2\ =\ ac$$    Eq.(i)

Let's assume the number must be added is 'y' to each of the numbers 1, 7 and 25 so that the resulting sums, in that order, maybe in continued proportion.

So a = (y+1)

b = (y+7)

c = (y+25)

Put the values of 'a', 'b' and 'c' in Eq.(i).

$$(y+7)^2 = (y+1)(y+25)$$

$$y^2+14y+49=y+25y+y^2+25$$

$$14y+49=26y+25$$

$$49-25=26y-14y$$

12y = 24

y = 2

So the required number which must be added is 2.