it takes 6 cubes to build a staircase with 3 steps. how many cubes will be needed for 11 steps?
1 step -- 1 cube
2 steps - 3 cubes
3 steps - 6 cubes
4 steps - 10 cubes
.....
n steps - n(n+1)/2 cubes
so 11 steps --11(12)/2 = 66 cubes
these are called the triangular numbers
think of it as the sum of billiard balls when they are racked up
1st row -- one ball
2 rows --- 3 balls
etc.
To determine the number of cubes needed for 11 steps, we can first analyze the pattern for the number of cubes required to build each step.
From the given information, we know that 6 cubes are needed for 3 steps. By observing the pattern, we can see that each step requires an additional 2 cubes. This means that for every step added, we need to add 2 cubes.
To find the number of cubes needed for 11 steps, we can use the formula:
Number of cubes = (Number of steps - 1) x 2 + Initial number of cubes
Plugging in the values, we get:
Number of cubes = (11 - 1) x 2 + 6
Number of cubes = 10 x 2 + 6
Number of cubes = 20 + 6
Number of cubes = 26
Therefore, to build a staircase with 11 steps, you would need 26 cubes.
To calculate the number of cubes needed for 11 steps, we can first find the pattern in the number of cubes required to build the stairs.
From the information given, we know that 6 cubes are required for 3 steps. We can see that each step takes 2 cubes to build. Therefore, we can deduce that each additional step will require an additional 2 cubes.
So, for 4 steps, we will need 6 + (2 * 1) = 8 cubes.
For 5 steps, we will need 6 + (2 * 2) = 10 cubes.
And so on, until we reach 11 steps.
Following this pattern, for 11 steps, we will need 6 + (2 * 8) = 22 cubes.