Let the sequence t_n be defined by t₁ = 1 and t_n = t_ n-1 + n for n 2 . The value of the sum _ n=1 ²⁰ (-1)^n t_n^2 is _____.

Step-by-step solution

From the given recurrence relation t_n - t_ n-1 = n with t₁ = 1 , we can recognize that t_n represents the sum of the first n natural numbers. Thus, t_n = n(n+1) 2 . We need to evaluate the sum S = _ n=1 ²⁰ (-1)^n t_n^2 . Writing out the terms, we get: S = -t₁^2 + t₂^2 - t₃^2 + t₄^2 - - t₁₉^2 + t₂₀^2 We can group the terms into 10 pairs of consecutive terms: S = _ m=1 ¹⁰ (t_ 2m ^2 - t_ 2m-1 ^2) Using the difference of squares identity, a^2 - b^2 = (a-b)(a+b) : t_ 2m ^2 - t_ 2m-1 ^2 = (t_ 2m - t_ 2m-1 )(t_ 2m + t_ 2