If sequence T is defined for all positive integers n such that t(n +1) = t(n) + n, and t 3 = 14, what is t 25?  Choose the correct option:
A) 101  B) 187  C) 191  D) 201  E) 311
ok, not sure if this is the best explanation, but here is my way of solving this :   tn +1 = tn + n,   t 3 = 14   so:   t 4 = t 3 + 3   t 5 = t 4 + 4, so we replace t 4 with the previous result : t 5 = (t 3 + 3 ) + 4   ....   so we get   t 25 = t 3 + 3 + 4 + 5 + 6....+ 24   the 3 + 4 + 5 + ... + 24 sequence is equal the sequence 1 + 2 + ... + 24, minus 1 + 2 at the beginning (so, minus 3)   and the 1 + 2 ... + 24 sequence is equal to n*(n+1)/2 with n being 24, so equal to  24 * 25 / 2= 
<gadget id="calculator"> 24 * 25 / 2</gadget>
<output>300</output>
300   so :   t 20= t 3 + 300 - 3   t 20 = 14 + 300 - 3 = 300 + 11= 
<gadget id="calculator">300 + 11</gadget>
<output>311</output>
311, hence answer 311
<result>E</result>
If sequence T is defined for all positive integers n such that t(n +1) = t(n) + n, and t 3 = 14, what is t 25?