A sum of ₹ 2,800 is to be used to award four prizes. If each prize after the first is ₹ 200 less than the preceding prize, find the value of each of these prizes.

Given,

Each prize is ₹ 200 less than the preceding prize.

Prizes : a, a - 200, a - 400, a - 600.

The above is an A.P., wth first term = a and common difference = ₹ -200.

Total prize money = ₹ 2,800.

By formula,

S_n = $\dfrac{n}{2}[2a + (n - 1)d]$

Substituting values we get :

$\Rightarrow 2800 = \dfrac{4}{2} \times [2 \times a + (4 - 1) \times -200] \\[1em] \Rightarrow 2800 = 2 \times [2a - 600] \\[1em] \Rightarrow 2800 = 4a - 1200 \\[1em] \Rightarrow 4a = 2800 + 1200 \\[1em] \Rightarrow 4a = 4000 \\[1em] \Rightarrow a = \dfrac{4000}{4} = ₹ 1,000.$

a - ₹ 200 = ₹ 1,000 - ₹ 200 = ₹ 800

a - ₹ 400 = ₹ 1,000 - ₹ 400 = ₹ 600

a - ₹ 600 = ₹ 1,000 - ₹ 600 = ₹ 400.

Hence, 1st Prize = ₹ 1,000, 2nd Prize = ₹ 800, 3rd Prize = ₹ 600, 4th Prize = ₹ 400.