The minimum age of children eligible to participate in a painting competition . . .

Question : The minimum age of children eligible to participate in a painting competition is 8 years. It is observed that the age of the youngest boy was 8 years and the ages of the participants, when seated in order of age, have a common difference of 4 months. If the sum of the ages of all the participants is 168 years, find the age of  the eldest participant in the painting competition. 

CBSE Board Exam 2025

Solution : 

Minimum age of children 
= 8 years
= 8×12 = 96 months

So, 

First Term of the AP (a) = 96 months
Common difference (d) = 4 months

Sum of the ages of all the participants (S_n)
= 168 years
= 168×12
= 2016 months

Let total number of participants be 'n'.

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

2016=n/2[2(96)+(n-1)(4)]
2016=n/2[192+4n-4]
2016=n/2[188+4n]
2016=[n/2]×2[94+2n]
2016=n[94+2n]
2016=94n+2n²
2n²+94n-2016=0
2[n²+47n-1008]=0
n²+47n-1008=0
n²+(63-16)n-1008=0
n²+63n-16n-1008=0
n(n+63)-16(n+63)=0
(n+63)(n-16)=0
n+63=0
n=-63
∵n≠-63
So n=-63 is rejected. 

n-16=0
n=16

So, there are 16 students. 

Now, 
a_n=a+(n-1)d
a16=96+(16-1)4
a16=96+15×4
a16=96+60
a16=156
=156 months
=156/12
= 13 years

So, age of eldest child is 156 months or 13 years.