Chapter 12.2, Problem 62AYU

Constructing a Brick Staircase A brick staircase has a total of 30 steps. The bottom step requires 100 bricks. Each successive step requires two fewer bricks than the prior step.

(a) How many bricks are required for the top step?

(b) How many bricks are required to build the staircase?

To determine

To find: A brick staircase has a total of 30 steps. The bottom step requires 100 bricks. Each successive step requires two fewer bricks than the prior step.

a. How many bricks are required for the top step?

Answer to Problem 62AYU

a. 42

Explanation of Solution

Given:

Total number of steps $=30$.

Number of bricks required for the bottom row $=100$.

Each successive step requires two fewer bricks than the prior step.

Formula used:

The $n$th term of an arithmetic sequence can be found by the formula,

$a_n=a_1+\left(n-1\right)d$   (Formula 1)

Sum of the first $n$ terms of an arithmetic sequence,

$S_n=\frac{n}{2}\left[a_1+a_n\right]$   (Formula 2)

Calculation:

From the given data, we get $n=30,a_1=\text{ 100},d=-2$.

a. To find the number of bricks are required for the top step :

$a_n=\text{ 100 }+\text{ 2}9\left(-2\right)=\text{ 42}$   (By formula 1)

So, the given sequence will be $100+98+96+\dots +42$

To determine

To find: A brick staircase has a total of 30 steps. The bottom step requires 100 bricks. Each successive step requires two fewer bricks than the prior step.

b. How many bricks are required to build the staircase?

Answer to Problem 62AYU

b. 2130

Explanation of Solution

Given:

Total number of steps $=30$.

Number of bricks required for the bottom row $=100$.

Each successive step requires two fewer bricks than the prior step.

Formula used:

The $n$th term of an arithmetic sequence can be found by the formula,

$a_n=a_1+\left(n-1\right)d$   (Formula 1)

Sum of the first $n$ terms of an arithmetic sequence,

$S_n=\frac{n}{2}\left[a_1+a_n\right]$   (Formula 2)

Calculation:

From the given data, we get $n=30,a_1=\text{ 100},d=-2$.

b. To find the number of bricks are required to build the staircase:

Therefore, the total number of bricks $=\frac{30}{2}\left[100+\text{ 42}\right]$

$=\text{ 15 }\times \text{ 142 }=\text{ 2130}$   (By formula 2)