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 prize.

Let the first prize be Rs. x. Thus each succeeding prize is Rs. 20 less than the preceding prize.

∴ Second prize, third prize, …, seventh prize be Rs. (x - 20), (x - 40) , …, (x - 120).


This forms an AP as x, x - 20, …, x - 120.


Here, first term = x


Common difference = x - 20 - x = - 20


Total number of terms = 7


Since, Total sum of prize amount = 700.


∴ Sum of all the terms = 700


Now, sum of first n terms of an AP is given by:


Sn = [2a + (n - 1)d]


∴ Sum of 7 terms of an AP is given by:


S7 = [2a + (7 - 1)d] = 700


⇒ [2x + (7 - 1)(-20)] = 700


⇒ 7[2x - 120] = 1400


⇒ 2x - 120 = 200


⇒ x - 60 = 100


⇒ x = 160


Thus, the prizes are as Rs. 160, Rs.140, Rs.120, Rs. 100, Rs. 80, Rs. 60, Rs. 40.


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