Let maximum house is 'n' ; sum of first 'n' even natural numbers =n2+n Let first 'm' even natural numbers are left in numbering the houses. (n2+n)−(m2+m)=170 ⇒n2−m2+n−m=170 ⇒(n−m)(n+m+1)=170 n−m=10⇒n−m=10 n+m+1=17⇒n+m=16 n=13;m=3 n≤13 If first term is a – 10 then sixth = a
n
2
[2(a−10)+2(n−1)]=170 ⇒n[a+n−11]=170 ⇒a=
170
n
+11−n ∴ n = 10 (only will make 'a' integer) ⇒a=17+11−10=18