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 of ages of all the participants is 168 years, find the age of eldest participant in the painting competition.
Given:
Age of youngest boy = 8 years
Common difference between ages = 4 months 
years
We can consider ages of the students in AP with
First term, a = 8 years
Common difference, 
Let the number of children be ‘n’.
As, sum of ages = 168 year
⇒ Sn = 168
Also, we know sum of n terms of an AP
Where,
a = first term
d = common difference
n = number of terms
Putting values, we get
⇒ Sn = 168
⇒ n(47 + n) = 1008
⇒ n2 + 47n – 1008 = 0
⇒ n2 + 63n – 16n – 1008 = 0
⇒ n(n + 63) – 16(n + 63) = 0
⇒ (n + 63)(n – 16) = 0
⇒ n = 16 or n = –63 [Not possible, as number of students can’t be negative]
Also, we know nth term of an AP
an = a + (n – 1)d
⇒ a16 
⇒ a16 = 13 years
Hence, age of eldest student is 13 years.
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