Four numbers are an arithmetic progression. If the sum of the numbers is 50 and the greatest number is four times the smallest number, then find the numbers.

Let the first number be a and common difference be d, Then arithmetic progression be a, a+d, a+2d, a+3d.

Given, a + a+d + a+2d + a+3d = 50
⇒ 4a + 6d = 50 …(i)
Also, a + 3d = 4(a)
⇒ 3d = 3a
⇒ a = d …(ii)

Solving eq. (i) and (ii), we get,
a = 5 and d = 5.

Hence the numbers are :
a₁ = 5
a₂ = a + d = 5 + 5 = 10
a₃ = a + 2d = 5 + 2(5) = 5 + 10 = 15
a₄ = a + 3d = 5 + 3(5) = 5 + 15 = 20.

Hence, The numbers are 5, 10, 15, 20.