In a cricket tournament 16 school teams participated. A sum of Rs 8000 is to be awarded among themselves as prize money. If the last placed team is awarded Rs 275 in prize money and the award increases by the same amount for successive finishing places, how much amount will the first place team receive?
Let the amount received by first place team be a Rs and d be difference in amount
As the difference is same hence the second-place team will receive a – d and the third place a – 2d and so on
The last team receives 275 Rs
As there are 16 teams and all teams are given prizes hence the sequence will have 16 terms because there are 16 teams
Therefore a, a – d, a – 2d… 275
The sequence is in AP with first term as a and common difference is ‘-d’
As the total prize given is of 8000rs hence
a + a – d + a – 2d … + 275 = 8000
The sum of n terms of AP is given by Sn = n/2 (2a + (n – 1) d) where
A is first term and d is common difference
There are 16 terms n = 16 and the sum Sn = 8000
⇒ 8000 = (16/2) (2a + (16 – 1) (-d))
⇒ 8000 = 8(2a – 15d)
⇒ 1000 = 2a – 15d ……1
The last term of AP is 275 and nth term of AP is tn = a + (n – 1) d
The last term is tn = 275
⇒ 275 = a + (16 – 1) (-d)
⇒ 275 = a – 15d ……2
Subtract 2 from 1 that is 1 – 2
⇒ 1000 – 275 = 2a – 15d – a + 15d
⇒ 725 = a
Hence amount received by first place team is Rs.725
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