A sum of ₹ 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is ₹ 20 less than its preceding prize, find the value of each of the prizes.

Solution:

Let the cost of the 1^st prize be ₹ P.

Cost of 2^nd prize = ₹ P − 20

And the cost of the 3^rd prize = ₹ P − 40

We can see that the cost of these prizes is in the form of AP, with a common difference of −20 and the first term as P.

Thus, a = P and d = −20

Given that, S ₇ = 700

By the formula of the sum of n terms, we know,

S_n = n /2 [2 a + ( n – 1) d ]

7/2 [2 a + (7 – 1) d ] = 700

2a + 6(-20) = 200

a + 3(−20) = 100

a −60 = 100

a = 160

Therefore, the value of each of the prizes was ₹ 160, ₹ 140, ₹ 120, ₹ 100, ₹ 80, ₹ 60, and ₹ 40.