The problem in your image shows a sequence of tile mosaics that grows in a specific pattern across stages. The number of tiles increases as the stages progress.

Observing the pattern:

• Stage 1: 1 tile.
• Stage 2: 4 tiles.
• Stage 3: 7 tiles.

It seems that each stage adds 3 more tiles compared to the previous one:

• From Stage 1 to Stage 2, the number of tiles increases by $4 - 1 = 3$.
• From Stage 2 to Stage 3, the number of tiles increases by $7 - 4 = 3$.

This suggests that the relationship between the stage number and the number of tiles follows an arithmetic sequence. The general formula for the number of tiles in an arithmetic sequence is:

$T(n) = a + (n - 1) \cdot d$

where:

• $a$ is the number of tiles at Stage 1 (which is 1),
• $d$ is the common difference (which is 3),
• $n$ is the stage number.

Formula for the nth stage:

$T(n) = 1 + (n - 1) \cdot 3 = 1 + 3n - 3 = 3n - 2$

So, the number of tiles at the nth stage is $T(n) = 3n - 2$.

Answers:

a. For the nth stage, the number of tiles is $3n - 2$.

b. For the 50th stage, substitute $n = 50$: $T(50) = 3(50) - 2 = 150 - 2 = 148$

So, 148 tiles are needed for the 50th stage.

Would you like further explanation on how the arithmetic sequence was determined? Here are some related questions:

1. How do arithmetic sequences apply to other growth patterns?
2. Can you explain the formula for the sum of an arithmetic sequence?
3. What happens if the common difference in a sequence changes?
4. How would the formula change if the pattern added more tiles per stage?
5. How can you graph the relationship between the stage number and the number of tiles?

Tip: Always look for a constant difference when analyzing patterns to identify arithmetic sequences.