(85pts)

1. Write a rule to find the number of blocks needed for the nth step. Explain your rule.

2. Write a rule to find the number of blocks needed to make a staircase with n number of steps. Explain your rule.

3. Write a rule that, given y number of blocks, you can use to determine how many steps are in the staircase. Explain your rule.

Step-by-step explanation:

1. The first step has 1 block.  The second step has 2 blocks.  The third step has 3 blocks.  So the nth step needs n blocks.

2. A staircase with n steps has 1 + 2 + 3 + ... + n steps.  This is an arithmetic sequence where the first term is 1 and the last term is n.  The sum of the first n terms is:

y = n/2 (1 + n)

3. Solve for n.

2y = n (1 + n)

2y = n² + n

0 = n² + n − 2y

Using quadratic formula:

n = [ -1 ± √(1 − 4(1)(-2y)) ] / 2

n = [ -1 + √(1 + 8y) ] / 2

1. The first step has 1 block.  The second step has 2 blocks.  The third step has 3 blocks.  So the nth step needs n blocks.

2. A staircase with n steps has 1 + 2 + 3 + ... + n steps.  This is an arithmetic sequence where the first term is 1 and the last term is n.  The sum of the first n terms is:

3. n = [ -1 + √(1 + 8y) ] / 2

Step-by-step explanation:

1. The first step has 1 block.  The second step has 2 blocks.  The third step has 3 blocks.  So the nth step needs n blocks.

2. A staircase with n steps has 1 + 2 + 3 + ... + n steps.  This is an arithmetic sequence where the first term is 1 and the last term is n.  The sum of the first n terms is:

y = n/2 (1 + n)

3. Solve for n.

2y = n (1 + n)

2y = n² + n

0 = n² + n − 2y

n = [ -1 ± √(1 − 4(1)(-2y)) ] / 2

n = [ -1 + √(1 + 8y) ] / 2