Question

If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are
(a) 5, 10, 15, 20
(b) 4, 10, 16, 22
(c) 3, 7, 11, 15
(d) none of these

Answer 1

(a) 5, 10, 15, 20

Let the four numbers in A.P. be as follows:
$a-2d,a-d,a,a+d$
Their sum = 50 (Given)
$$\begin{gathered} \Rightarrow \left(a-2d\right)+\left(a-d\right)+a+\left(a+d\right)=50 \\ \Rightarrow 2a-d=25.....\left(1\right) \\ \mathrm{Also},\left(a+d\right)=4\left(a-2d\right) \\ \Rightarrow a+d=4a-8d \\ \Rightarrow 3d=a......\left(2\right) \end{gathered}$$

From equations $\left(1\right)\mathrm{and}\left(2\right),$ we get:
d = 5, a = 15

Hence, the numbers are $15-10,15-5,15,15+5$, i.e. 5, 10, 15, 20.