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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
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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