Please help solve the below word problem

Question

1 step stair uses 1 block 2 step stairs uses 3 blocks 3 step stair uses 6 blocks how many does 10 step stair use? how many blocks for a 37 step stair? how many blocks will be needed to build a n-step?

Need help finding the formula to compute the above thank you!

Answer 1

# stairs (n)        # blocks

1                       1

2                       1 +2 = 3

3                       1 + 2 + 3 = 6

4                       1 + 2 + 3 +4 = 10

 

For n = 4, think of 4 blocks as the base, with 3 blocks on top of them, 2 blocks on top of the 3, and finally 1 block on top of the 2. You are actually building a pyramid to form the stairs.

n = 10   the number of blocks = 10 x 11/2 =55

n = 37   the number of blocks = 37x38/2 = 703

 

For any number of stairs n, the total number of blocks = n(n+1)/2

Answer 2

The number of step stairs depends on the number of blocks used.  This is because without blocks, you cannot build stairs.

 

Let x = number of blocks

Let y = number of step stairs

 

Create a table of values to show the data.

 

x      |      y

__________

 

1            1

3            2

6            3

 

Lets see if this data has a linear relationship.  We do this by find the slope.  The slope between points 1 and 2 must have the slope between points 2 and 3.

 

Slope_1-3 = (2 - 1) / (3 - 1)

             = 1 / 2

 

Slope_2-3 = (3 - 2) / (6 - 3)

            = 1 / 3

 

The slope is not constant.  So we know we won't have a linear relation.

 

So now we need to write a system of linear equations to find a single equation.  The equation we are trying to get is a quadratic function.  The function will be in the form

 

y = ax² + bx + c

 

where a, b, and c are constants.  We plug in the x and y values from the table to create these equations.

 

a + b + c = 1                   eq1

9a + 3b + c = 2               eq2

36a + 6b + c = 3             eq3

 

 

Substitute eq1 into eq2 and eq3.  Get these equations in terms of a and b.

 

9a + 3b + (1 - a - b) = 2

8a + 2b = 1                        new eq2

 

36a + 6b + (1 - a - b) = 3

35a + 5b = 2                       new eq3

 

 

Next, multiply new eq2 by 5 and multiply new eq3 by 2.

 

40a + 10b = 5            new eq2

70a + 10b = 4            new eq3

 

 

Next, subtract new eq3 from new eq2 to eliminate the b terms.

 

-30a = 1

 

    a = -1/30

 

 

Substitute this value of a into new eq2 to solve for b.

 

8(-1/30) + 2b = 1

 

(-8/30) + 2b = 1

 

-8 + 60b = 30

 

       60b = 38

 

           b = 19/30

 

 

Plug in the values of a and b into eq1 to solve for c.

 

c = 1 - (-1/30) - (19/30)

 

c = 2/5

 

 

The formula that we get when we plug in a, b, and c is

 

y = (-1/30)x² + (19/30)x + (2/5)