Hello, nrmegin!

It helps if you know this formula . . .

The sum of integers from 1 to is given by:

. .

A landscape architect needs to design a path for the Museum of Modern Art.

The path is to be made of black and red tiles in the following pattern:

1 black, 1 red, 2 black, 2 red, 3 black, 3 red, 4 black, 4 red, ...

The pattern of alternating colors is to continue so that each block of a single color

has one more tile than the preceding block of the same color.

The museum curator estimated that 5000 tiles will be needed for the project.

She ordered 2500 black tiles and 2500 red ones.

If the landscape architect wants to use all of these tiles and have the path end

with a block of black tiles followed by the same number of red tiles,

what is the smallest number of additional black and red tiles that should be ordered?

The total number of tiles is: .

. .

If , we have: .

Quadratic Formula: .

. . The positive root is: .

If , then: .

. . Therefore, a minimum of 112 more tiles are needed.