An architect is designing a building in the shape of a rectangular pyramid. The number of windows on each floor forms an arithmetic sequence. There are 76 windows on the 7th floor, 68 on the 8th floor, and 4 on the very last floor
Find an explicit formula to show this sequence
Find a recursive formula to show this sequence
How tall is the building (ground floor is the first floor)
On the nth floor
T_n = 76-4((n-7)) = 104-4n
T_7 = 76
T_n+1 = T_n - 4
(76-4)/4 + 1 = 19 rows starting at the 7th, making 26 floors in all.
To find an explicit formula for the arithmetic sequence of windows, we need to find the common difference and the first term.
First, let's find the common difference (d) between successive terms:
d = a2 - a1
= 68 - 76
= -8
Here, a1 represents the number of windows on the 7th floor and a2 represents the number of windows on the 8th floor.
Now, let's find the first term (a) of the sequence:
a = a1 - (n - 1)d
= 76 - (7 - 1)(-8)
= 76 - 6(-8)
= 76 + 48
= 124
Therefore, the explicit formula for the arithmetic sequence is:
an = 124 - 8(n - 1)
To find the recursive formula, we need to determine the difference between successive terms. Since we have the common difference, d = -8, the recursive formula can be represented as follows:
an = an-1 + d
For the first term, we have:
a1 = 76
Thus, the recursive formula for the sequence is:
an = an-1 - 8
To determine the height of the building, we need to find the number of floors.
We know that there are 4 windows on the very last floor, so this represents the final term of the sequence, a(n). Using the explicit formula, we can solve for n:
4 = 124 - 8(n - 1)
Rearranging the equation, we get:
8(n - 1) = 124 - 4
8(n - 1) = 120
n - 1 = 120/8
n - 1 = 15
n = 16
Therefore, the ground floor (first floor) is included in the total number of floors, so the building is 16 floors tall.