Alicia wants to build a block pyramid similar to the one shown. block pyramid the top level will always have 2 blocks. Alicia wants her pyramid to contain as many levels as possible. How many levels can her pyramid have if Alicia has 200 blocks?

To determine the number of levels Alicia's pyramid can have, we first need to understand the pattern of blocks in each level.

In the given pyramid, the top level has 2 blocks. Each subsequent level below has an additional row, with one block less on each end compared to the level above it. For example, the second level has two blocks on each end and one block in the middle, totaling 5 blocks.

To calculate the number of blocks in each level, we can use the formula:
Number of blocks = (Number of blocks in the first row) + (Number of blocks in the last row)
For example, for the second level: Number of blocks = 2 + 3 = 5

Looking at the pyramid, we can observe that the number of blocks in each level forms an arithmetic sequence, increasing by 3 each time. So, to find the total number of blocks for a given level, we can use the formula for the sum of an arithmetic sequence.

Sum of an arithmetic sequence = n/2 * (first term + last term)
where n is the number of terms in the sequence.

In this case, n represents the number of levels, and the number of blocks in the first and last rows will be 2 and 2 + ((n-1) * 3), respectively.

So, the sum of the arithmetic sequence (total number of blocks) is given by:
200 = n/2 * (2 + 2 + (n-1) * 3)

Now, we can solve this equation to find the value of n, which represents the maximum number of levels Alicia's pyramid can have.

Let's perform the calculations:
200 = n/2 * (4 + 3n - 3)

Multiplying both sides by 2 to eliminate the fraction:
400 = n * (3n + 1)

Rearranging the equation:
3n^2 + n - 400 = 0

This is a quadratic equation, so we can solve for n using the quadratic formula:
n = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 3, b = 1, and c = -400.

Calculating the quadratic formula:
n = (-1 ± √(1^2 - 4 * 3 * -400)) / (2 * 3)
n = (-1 ± √(1 + 4800)) / 6
n = (-1 ± √4801) / 6

The discriminant (√4801) is positive, so we have two possible solutions:
n = (-1 + √4801) / 6 ≈ 11.29 or n = (-1 - √4801) / 6 ≈ - 1.96

Since the number of levels must be a positive integer, the only valid solution is:
n ≈ 11.29 (rounded down to the nearest whole number) = 11

Therefore, Alicia's pyramid can have a maximum of 11 levels if she has 200 blocks.

depends on the shape of the pyramid...