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Question
Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.
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Solution
\[\text { Let the four numbers be }a - 3d, a - d, a + d, a + 3d . \]
\[\text { Sum } = 50\]
\[ \Rightarrow a - 3d + a - d + a + d + a + 3d = 50\]
\[ \Rightarrow 4a = 50\]
\[ \Rightarrow a = \frac{25}{2} . . . (i)\]
\[\text { Also }, a + 3d = 4(a - 3d)\]
\[ \Rightarrow a + 3d = 4a - 12d\]
\[ \Rightarrow 3a = 15d\]
\[ \Rightarrow a = 5d\]
\[ \Rightarrow \frac{25}{2 \times 5} = d \left( \text { Using } (i) \right)\]
\[ \Rightarrow \frac{5}{2} = d\]
\[\text { So, the terms are as follows: } \]
\[ \left( \frac{25}{2} - 3 \times \frac{5}{2} \right), \left( \frac{25}{2} - \frac{5}{2} \right), \left( \frac{25}{2} + \frac{5}{2} \right), \left( \frac{25}{2} + 3 \times \frac{5}{2} \right)\]
\[ = 5, 10, 15, 20\]
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