Question

A brick staircase has a total of 25 steps. The bottom step requires 80 bricks. Each

successive step requires three 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?

Short answer

(a) Number of bricks required for the top step are $8.$

(b) Total number of bricks required to build the staircase are $1100.$

Step-by-step solution

Given information:

Total number of steps, $n=25$,

Number of bricks for bottom step, $a_1=80$,

Since each successive step requires three less bricks, $d=-3$

$\therefore$Sequence will be $80,77,74,...$. It is an Arithmetic Sequence.

Formulas: 

$$\begin{gathered} \left(1\right)a_n=a_1+(n-1)d \\ \left(2\right)S_n=\frac{n}{2}\left(a_n+a_1\right) \end{gathered}$$

(a) To find the number of bricks required for the top step ie, an:

Substitute the given values in the first formula, we get,

$a_n=80+(25-1)\times \left(-3\right)$

$=80+24\times \left(-3\right)$

$=80-72$

$=8$

(b) To find the number of bricks to build the staircase ie, Sn.

Substitute the given values in the second formula, we get,

$S_n=\frac{25}{2}\left(80+8\right)$

$$\begin{gathered} =\frac{25}{2}\times 88 \\ =25\times 44 \end{gathered}$$

$=1100$