The image you uploaded shows a problem asking how many tiles are needed for the 100th row, based on a pattern shown in the figure.

Observing the Pattern:

1. In Row 1, there is 1 tile.
2. In Row 2, there are 3 tiles.
3. In Row 3, there are 5 tiles.

The pattern is increasing by 2 tiles per row:

• Row 1: 1 tile
• Row 2: 3 tiles (1 + 2)
• Row 3: 5 tiles (3 + 2)
• Row 4: 7 tiles (5 + 2), and so on.

This is an arithmetic sequence where:

• The first term $a_1 = 1$ (number of tiles in Row 1),
• The common difference $d = 2$.

Formula for the $n$-th term of an arithmetic sequence:

$a_n = a_1 + (n - 1) \cdot d$ Where:

• $a_n$ is the number of tiles in the $n$-th row,
• $n$ is the row number,
• $d$ is the common difference (2 in this case).

For the 100th row ($n = 100$): $a_{100} = 1 + (100 - 1) \cdot 2 = 1 + 99 \cdot 2 = 1 + 198 = 199$

So, Estella will need 199 tiles for the 100th row.

Would you like more details or have any questions?

Related Questions:

1. What is the total number of tiles needed for the first 100 rows?
2. How would the pattern change if the difference between rows was 3 tiles instead of 2?
3. What is the general formula for the sum of an arithmetic sequence?
4. How many tiles would be needed for the 50th row?
5. What if the pattern had started with 2 tiles in the first row instead of 1?

Tip: Always identify the type of sequence (arithmetic, geometric, etc.) when analyzing a pattern to apply the correct formula.