Triangle Numbers

Image the triangle for $T_{100}$ , with 100 dots in the bottom row. Now take the triangle for $T_{99}$ , turn it upside down, and place it underneath the first triangle. The resulting figure is a diamond-- a square standing on one of its corners. Since the sides of the square contain 100 dots, the total number of dots is $T_{100} + T_{99} = 100^2 = \boxed{10\:000}.$

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Alternatively, it is not difficult to show that $T_n = \tfrac12 n (n+1).$ We find $T_{100} + T_{99} = \tfrac12\cdot 100\cdot 101 + \tfrac12\cdot 99 \cdot 100 \\ = \tfrac12\cdot 100 \cdot (101 + 99) = \tfrac12\cdot 100\cdot 200 = \boxed{10\:000}.$

From the illustration above, it is clear that $T_{n-1}+T_n = n^2 \implies T_{99}+T_{100}=100^2 = \boxed{10000}$ .

$\frac{n(n+1)}{2}$

Use from $\uparrow$ for find $T_{99} + T_{100}$ .

For $T_{99}$ we have $\frac{99(99+1=100)}{2} = 4950$

For $\text{T}_{100}$ we have $\frac{100(100+1=101)}{2} = 5050$

For $T_{99} + T_{100}$ we have $\text{(4950+5050) = 10000}$

$\therefore$ the answer is $\boxed{\color{#302B94}{10000}}$

We can find a pattern here,

T2 = (2+1)C(2-1) = 3

T3 = (3+1)C(3-1) = 6

T4 = (4+1)C(4-1) = 10 ...

...

Tn = (n+1)C(n-1)

Hence, T99 + T100 = (99+1)C(99-1) + (100+1)C(100-1)

. = 100 * 99/2 + 101 * 100/2 .
. = 50 * 99 + 101 * 50

. = 50(99+101)

. = 10000