A finite set M of positive integers consists of distinct perfect squares and the number 92 . The average of the numbers in M is 85 . If we remove 92 from M , the average drops to 84 . If N^2 is the largest possible square in M , what is the value of N ?

Step-by-step solution

a₁^2+a₂^2+a₃^2+ +a₀^2+92 n+1 =85 ......(i) If we remove 92 , then a₁^2+a₂^2+a₃^2+ +a_n^2 n =84 ......(ii) From (i) and (2) aligned & 84 n+92 85 n+85 & n=7 aligned Now, a₁^2+a₂^2+a₃^2+ +a₆^2+a₇^2=84 7=588 If a₁=1, a₂=2, a₃=3, a₄=4, a₅=5, a₆=6 Then 1^2+2^2+3^2+4^2+5^2+6^2+a₇^2=588 a₇^2=588-91 a₇^2=497 (not possible) If a₁=1, a₂=2, a₃=3, a₄=4, a₅=5, a₆=7 Then 1^2+2^2+3^2+4^2+5^2+7^2+a₇^2=588 array ll & a₇^2=484 & a₇=22 & N=22 array