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

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

(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 in a group of six, always ending with $\square$, both the 1st and the $166^{\circ}$ group. Since $996=6 \times 166$, the last figure of the $166^{\text{th}}$ group, that is, the 996th figure, is $\square$.

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-067.jpg?height=166&width=1211&top_left_y=1776&top_left_x=457)

(a) The $1000^{\text{th}}$ figure, therefore, is

(b) The first $\diamond$ is in the $3^{\text{rd}}$ position, the second is in the $1 \times 6+3=9^{\text{th}}$ position, the third is in the $2 \times 6+3=15^{\text{th}}$ position, the fourth is in the $3 \times 6+3=21^{\text{st}}$ position, and so on, up to the thousandth $\diamond$, which appears in the $999 \times 6+3=5997^{\text{th}}$ position.