The figures $\triangle, \boldsymbol{\Lambda}, \diamond, \uparrow, \odot, \square$ are repeated in the sequence

$$
\triangle, \boldsymbol{\phi}, \diamond, \boldsymbol{\phi}, \odot, \square, \triangle, \boldsymbol{\phi}, \diamond, \boldsymbol{\phi}, \odot, \square, \ldots
$$

(a) Which figure will appear in the $1000^{\text{th}}$ position of the sequence?

(b) In which position does the thousandth $\diamond$ appear?
The figures repeat every 6. Dividing 1000 by 6 we get: $1000=6 \times 166+4$.

![](https://cdn.mathpix.com/cropped/2024_05_01_7802a0a2c42db868808eg-6.jpg?height=186&width=1219&top_left_y=861&top_left_x=521)

(a) The figure that is in the $1000^{\text{th}}$ place is

(b) The first $\diamond$ is in the $3^{\text{rd}}$ position, the second is in the position numbered $3+6$, the third in $3+6+6$, and so on, as indicated below: $\mathbf{1}^{\text{st}} \quad \longrightarrow \quad 3+\mathbf{0} \times 6$ $\mathbf{2}^{\text{nd}} \quad \longrightarrow \quad 3+\mathbf{1} \times 6$ $\mathbf{3}^{\text{rd}} \longrightarrow 3+\mathbf{2} \times 6$ $4^{\text{th}} \longrightarrow 3+\mathbf{3} \times 6$ $1000^{\text{th}} \longrightarrow 3+\mathbf{999} \times 6$

Therefore, the $1000^{\text{th}} \diamond$ appears in the position: $3+999 \times 6=5997$.