Olympiad workbookIOQMSequences and Series

Let m be the smallest positive integer such that m^2+(m+1)^2+ +(m+10)^2 is the square of a positive integer n . Find m+n .

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aligned _ r=1 ¹¹(m+r-1)^2 & =11 m^2+ (1^2+2^2+3^2+ +10^2 )+2 m(1+2+3+ +10) & =11 (m^2+10 m+35 ) & =11 ((m+5)^2+10 ) aligned The least possible value of m=18 The required sum =11 (23^2+10 ) aligned & =11 11 49 & =77^2 aligned Then, m=18 and n=77 m+n=95

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