SOLUTION: The windows of a downtown office building are arranged so that each floor has six fewer windows than the floor below it. If the ground floor has 52 windows, how many total windows

Click here to see ALL problems on Sequences-and-series

Question 1204026: The windows of a downtown office building are arranged so that each floor has six fewer windows than the floor below it. If the ground floor has 52 windows, how many total windows are on the first eight floors
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52787)   (Show Source):
You can put this solution on YOUR website!
.
The windows of a downtown office building are arranged so that each floor
has six fewer windows than the floor below it. If the ground floor has 52 windows,
how many total windows are on the first eight floors
~~~~~~~~~~~~~~~~~~~~

First floor has 52 windows. Eighth floor has 52 - (8-1)*6 = 52 - 7*6 = 52 - 42 = 10 windows (use the formula for for the n-th term of an arithmetic progression). The total windows is the sum of an arithmetic progression W = = 248 ANSWER (use the formula for the sum of an arithmetic progression W = .

Solved, with explanations.

------------------

For introductory lessons on arithmetic progressions see
    - Arithmetic progressions
    - The proofs of the formulas for arithmetic progressions
    - Problems on arithmetic progressions
    - Word problems on arithmetic progressions
in this site.

Find there everything yo need, and
learn from there everything you need.

Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!

Answer: 248 windows

Explanation

In some countries, the 1st floor is considered to be above the ground floor.
However, I'll consider the ground floor to be the 1st floor.

Here's the long way to do it without needing any formulas
1st floor = 52 windows
2nd floor = 52-6 = 46 windows
3rd floor = 46-6 = 40 windows
4th floor = 40-6 = 34 windows
5th floor = 34-6 = 28 windows
6th floor = 28-6 = 22 windows
7th floor = 22-6 = 16 windows
8th floor = 16-6 = 10 windows
In total there are 52+46+40+34+28+22+16+10 = 248 windows

Typing all that into a calculator is fairly tedious. Luckily there's a faster method.
The sequence 52,46,34,... is arithmetic with
a1 = 52 = first term
d = -6 = common difference
From there we would write:
Sn = sum of the first n terms
Sn = (n/2)*(2*a1 + d*(n-1))
S8 = (8/2)*(2*52 - 6*(8-1))
S8 = 248

Here's another method.
The nth term of an arithmetic sequence is: an = a1 + d(n-1)
Plug in a1 = 52 and d = -6, and simplify, to get an = -6n+58
Plug in n = 8 to get a8 = 10 to show there are 10 windows on the 8th floor.
Then,
Sn = sum of the first n terms
Sn = (n/2)*(first + last)
Sn = (n/2)*(a1 + an)
S8 = (8/2)*(a1 + a8)
S8 = (8/2)*(52 + 10)
S8 = 248