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

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 where

A is first term and d is common difference

There are 16 terms n = 16 and the sum S_n = 8000

⇒ 8000 = 8(2a – 15d)

⇒ 1000 = 2a – 15d …(i)

The last term of AP is 275 and n^th term of AP is t_n = a + (n – 1)d

The last term is t_n = 275

⇒ 275 = a + (16 – 1)(-d)

⇒ 275 = a – 15d …(ii)

Subtract (ii) from (i) that is (i) – (ii)

⇒ 1000 – 275 = 2a – 15d – a + 15d

⇒ 725 = a

Hence amount received by first place team is 725 Rs.