How much must be added to each of the three number 1, 11, and 23 so that together they form a geometric progression?

49

Step-by-step explanation:

Let the three numbers be a1, a2, a3

Let the number to be added to form a geometric progression = x

a1 = 1 + x ; a2 = 11 + x ; a3 = 23 + x

For a geometric progression :

a2 = a1 * r

a2 = (1+x)r

11+x = (1+x)r - - (1)

r = (11+x) / (1+x) ---(1)

a3 = a2 * r

a3 = (11+x)r

23+x = (11+x)r - - -

r = (23+x) / (11+x) - - (2)

Equate (1) and (2)

(11+x) / (1+x) = (23+x) / (11+x)

(11+x) * (11+x) = (23+x) * (1+x)

121 + 22x + x² = 23 + 24x + x²

121 + 22x + x² - x² - 24x - 23 = 0

98 - 2x = 0

-2x = - 98

x = 98 / 2

x = 49

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