Triangle Numbers

Algebra Level 3

To construct the nnth triangle number, start with a row of nn dots. Then add a row with n1n-1 dots. Continue, adding a row with one fewer dot in every step, until a row with one dot is added. The drawing above show the second through sixth triangle numbers: T2=3,T3=6,T4=10,T5=15,T6=21T_2 = 3, T_3 = 6, T_4 = 10, T_5 = 15, T_6 = 21.

How much is T99+T100T_{99} + T_{100}?

Try to answer without a calculator.

Hint: Algebraic answers are great; geometric answers may be even greater!


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