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

Let the cost of 1st prize be P Rs.
Cost of 2nd prize = P - 20 Rs
And cost of 3rd prize = P - 40 Rs

We can see that the cost of these prizes are in the form of A.P., having common difference as -20 and first term as P.

Thus, a = P and d = -20
Given that, \(S_7 = 700\)

By the formula of sum of nth term, we know,
\(\Rightarrow S_n = {{n} \over {2}} [2a + (n – 1)d]\)
\(\Rightarrow{{7} \over {2}} [2a + (7 – 1)d] = 700\)
\(\Rightarrow{{2a + (6)(-20)} \over {2}} = 100\)
\(\Rightarrow \) a + 3(-20) = 100
\(\Rightarrow \) a - 60 = 100
\(\Rightarrow \) a = 160

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