Question

The minimum age of children to be eligible to participate in a painting competition is 8 years. It is observed that the age of youngest boy was 8 years and the ages of rest of participants are having a common difference of 4 months. If the sum ages of all the participants is 168 years. Find the age of eldest participant in the painting competition.

Answer 1

common difference d = 4 months = 4/12 year = 1/3
AP = 8, 8+1/3, 8+1/3 +1/3............till the eldest participant
a=8
d=1/3
S_n = 168
n/2[2a + (n-1)d]=168
n/2[2(8) + (n-1)(1/3)]=168
n[16 + n/3-1/3]= 168 *2
16n +n²/3 - n/3=336

48n +n² - n = 1008
n² +47n -1008=0
n² +63n -16n -1008=0
n(n+63)-16(n+63)=0
(n-16)(n+63)=0
n-16=0 or n+63=0
n=16 or n = -63
there fore no. of participants are 16

eldest participant = a_n
= a + (n-1)d
=8 +(15)(1/3)
=8+5
=13 years