Let be the smallest positive integer such that m2 + (m + 1)2 + … + (m + 10)2 is the square of positive integer n. Find + n.
Hint 1: Start by analyzing the initial conditions and setting up the basic equations. (m 1)2 11m 2 (12 22 32 \dots 102 ) 2m(1 2 3 \dots 10).
Hint 2: Look for algebraic properties, symmetry, or geometric theorems to simplify. = 11(m2 + 10m + 35).
Hint 3: Proceed with the final algebraic steps to solve the system. solve for the final value((m + 5)2 + 10).
Step 1: (m 1)2 11m 2 (12 22 32 \dots 102 ) 2m(1 2 3 \dots 10)
Step 2: = 11(m2 + 10m + 35)
Step 3: = 11((m + 5)2 + 10)
Step 4: The least possible value of = 18
Step 5: The required sum = 11(232 + 10)
Step 6: = 11 × 11 × 49
Step 7: Then, = 18 and = 77
Step 8: => + = 95
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