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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
Options
5, 10, 15, 20
4, 10, 16, 22
3, 7, 11, 15
none of these
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Solution
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)
\[\Rightarrow \left( a - 2d \right) + \left( a - d \right) + a + \left( a + d \right) = 50\]
\[ \Rightarrow 2a - d = 25 . . . . . \left( 1 \right)\]
\[\text { Also }, \left( a + d \right) = 4\left( a - 2d \right)\]
\[ \Rightarrow a + d = 4a - 8d\]
\[ \Rightarrow 3d = a . . . . . . \left( 2 \right)\]
From equations \[\left( 1 \right) \text { and }\left( 2 \right),\] , we get:
d = 5, a = 15
Hence, the numbers are
\[15 - 10, 15 - 5, 15, 15 + 5\], i.e. 5, 10, 15, 20.
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