If the sum of the first N terms of the series 1^2 - 3^2 + 5^2 - 7^2 + is equal to 3361 , then the value of N is

Step-by-step solution

Since the sum 3361 is positive and the terms of the series are increasing in magnitude with alternating signs, the last term must be positive. This means N must be an odd number. Let N = 2m+1 . We group the first 2m terms into pairs: Sum = (1^2 - 3^2) + (5^2 - 7^2) + + ((4m-3)^2 - (4m-1)^2) + (4m+1)^2 Using a^2 - b^2 = (a-b)(a+b) , each pair simplifies to: (4k-3)^2 - (4k-1)^2 = (-2)(8k-4) = -16k + 8 For k=1 , the pair is -8 . For k=2 , the pair is -24 . For k=m , the pair is -16m + 8 . The sum of these m pairs form