Let a sequence T_n be defined as T_n = 1 2 3 + 2 3 4 + + n(n+1)(n+2) 1 2 + 2 3 + + n(n+1) . The value of _ n=1 ²⁵ T_n is

Step-by-step solution

The numerator of T_n is the sum of products of three consecutive integers: _ k=1 ^n k(k+1)(k+2) = n(n+1)(n+2)(n+3) 4 The denominator of T_n is the sum of products of two consecutive integers: _ k=1 ^n k(k+1) = n(n+1)(n+2) 3 Substituting these into the expression for T_n : T_n = n(n+1)(n+2)(n+3) 4 n(n+1)(n+2) 3 T_n = 3 4 (n+3) We need to find the sum of the first 25 terms of this sequence: _ n=1 ²⁵ T_n = _ n=1 ²⁵ 3 4 (n+3) = 3 4 _ n=1 ²⁵ (n+3) The sum _ n=1 ²⁵ (n+3) is an arithmetic progression with 25 terms, first