Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.

Question:

Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.

Solution:

Let the four numbers be $a-3 d, a-d, a+d, a+3 d$.

Sum $=50$

$\Rightarrow a-3 d+a-d+a+d+a+3 d=50$

$\Rightarrow 4 a=50$

$\Rightarrow a=\frac{25}{2} \ldots(i)$

Also, $a+3 d=4(a-3 d)$

$\Rightarrow a+3 d=4 a-12 d$

$\Rightarrow 3 a=15 d$

$\Rightarrow a=5 d$

$\Rightarrow \frac{25}{2 \times 5}=d \quad(\operatorname{Using}(\mathrm{i}))$

$\Rightarrow \frac{5}{2}=d$

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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