https://file.aiquickdraw.com/mathbot/file/1726789055724k46px9b8.jpg

The problem involves a growing "staircase" sequence, where each step in the sequence adds another row of cubes. Based on the image, the pattern starts with:

1. The first step having 1 cube.
2. The second step having 3 cubes (2 cubes stacked on the previous 1).
3. The third step having 6 cubes (3 cubes stacked on the previous 3).
   
This pattern can be described as triangular numbers. The number of cubes needed to build the \(n\)-th staircase is the sum of the first \(n\) natural numbers:

\[
T_n = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}
\]

Now, we need to predict the number of cubes in the 50th staircase:

\[
T_{50} = \frac{50(50 + 1)}{2} = \frac{50 \times 51}{2} = 1275
\]

So, the total number of cubes required to build the 50th staircase is **1275**.

Would you like more details or have any questions? Here are some related questions to explore:

1. How would the formula change if the pattern added two cubes per step instead of one?
2. Can you find a recursive relation for this staircase sequence?
3. How does this relate to arithmetic sequences and series?
4. What is the sum of cubes needed for the first 100 staircases?
5. Can you find a geometric representation of this problem?

**Tip**: The formula for the sum of the first \(n\) natural numbers, \( \frac{n(n+1)}{2} \), is a key tool in understanding many similar growing patterns.

This 'staircase' sequence shows a growing pattern. Predict the total number of cubes you will need to build the 50th staircase. Show your thinking.

Explore how to predict the total number of cubes needed to build the 50th staircase in a growing staircase sequence. Using triangular numbers and the summation formula n(n + 1)/2, we calculate that 1275 cubes are needed. Suitable for students in Grades 6-8.

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