# Thread: arithmetic series

1. ## arithmetic series

q:
Question: A set of steps for the end of a pier are built of stone. A sketch of the cross-section of the steps is shown.

Each step has rise of 0.2m and tread of 0.6m. Form a series to calculate the area of the cross-section.

I tried to answer the question but I got it wrong
I tried to to find the area of A
using Sums of arithmetic - i found the length of the stairs
$\displaystyle S_{12}= \frac{12}{2}\left [ 2(0.6)+(12-1)0.2 \right ] = 19.8$
area of a = $\displaystyle \frac{1}{2}(19.8)(12) = 118.8$
area b = 8 x 12 = 96

therefore total area = 214.8 square metres

the answer in the book is 118.8 square metres

Can someone please help

2. ## Re: arithmetic series

12/0.6 = 20 rectangular strips of width 0.6 m and variable height 8+0.2n meters

$S_{20}=(8 \cdot 0.6) + [(8+0.2) \cdot 0.6] + [(8+0.4) \cdot 0.6] + ... + [(8+3.8) \cdot 0.6]$

$S_{20} = 0.6(8+8.2+8.4+ ... +11.8) = 0.6\bigg[\dfrac{20}{2}(16+19 \cdot 0.2) \bigg] = 118.8 \, m^2$

3. ## Re: arithmetic series

In case this helps:

h = height = 8
w = width = 12
So area without steps = hw = 96

r = rise = .2
t = tread = .6

Arithmetic series:
S = n[2a + d(n - 1)]/2
a = 1st term = rt
d = difference = rt (so a=d)
n = number of terms = w/t - 1

Area = hw + n[2a + d(n - 1)]/2 = 118.8 (as per Skeeter)

NOTE:
don't ask me to explain further: no blackboard