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: Option AShow Answer
Solution(By Apex Team)
4 numbers are in A.P.
Let the numbers be
a – 3d, a – d, a + d, a + 3d
Where a is the first term and 2d is the common difference
Now their sum = 50
a – 3d + a – d + a + d + a + 3d = 50
and greatest number is 4 times the least number
a + 3d = 4 (a – 3d)
a + 3d = 4a – 12d
4a – a = 3d + 12d
⇒ 3a = 15d
$\begin{aligned}&\Rightarrow a=\frac{15d}{3}=5d\\
&\Rightarrow\frac{25}{2}=5d\\
&\Rightarrow d=\frac{25}{2\times5}\\
&\Rightarrow d=\frac{5}{2}\end{aligned}$
$\begin{aligned}&\therefore\text{ Numbers are }\\
&\frac{25}{2}-3\times\frac{5}{2},\ \frac{25}{2}-\frac{5}{2},\ \frac{25}{2}+\frac{5}{2},\ \frac{25}{2}+3\times\frac{5}{2}\\
&\Rightarrow\frac{10}{2},\ \frac{20}{2},\ \frac{30}{2},\ \frac{40}{2}\\
&\Rightarrow5,\ 10,\ 15,\ 20\end{aligned}$
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