Complete Question:
There are 8 rows and 8 columns, or 64 squares on a chessboard.
Suppose you place 1 penny on Row 1 Column A,
2 pennies on Row 1 Column B,
4 pennies on Row 1 Column C, and so on …
Determine the number of pennies in Row 1
Determine the number of pennies on the entire chessboard?
Answer:
255 in the first row
18,446,744,073,709,551,615 in the entire board
Step-by-step explanation:
Given
![Rows = 8](https://tex.z-dn.net/?f=Rows%20%3D%208)
![Columns = 8](https://tex.z-dn.net/?f=Columns%20%3D%208)
Solving (a): Number of pennies in first row
The question is an illustration of geometric sequence which follows
![1,2,4....](https://tex.z-dn.net/?f=1%2C2%2C4....)
Where
--- The first term
Calculate the common ratio, r
![r = \frac{T_2}{T_1} = \frac{4}{2} = 2](https://tex.z-dn.net/?f=r%20%3D%20%5Cfrac%7BT_2%7D%7BT_1%7D%20%3D%20%5Cfrac%7B4%7D%7B2%7D%20%3D%202)
The number of pennies in the first row will be calculated using sum of n terms of a GP.
![S_n = \frac{a(r^n - 1)}{n - 1}](https://tex.z-dn.net/?f=S_n%20%3D%20%5Cfrac%7Ba%28r%5En%20-%201%29%7D%7Bn%20-%201%7D)
Since, the first row has 8 columns, then
![n = 8](https://tex.z-dn.net/?f=n%20%3D%208)
Substitute 8 for n, 2 for r and 1 for a in ![S_n = \frac{a(r^n - 1)}{r - 1}](https://tex.z-dn.net/?f=S_n%20%3D%20%5Cfrac%7Ba%28r%5En%20-%201%29%7D%7Br%20-%201%7D)
![S_8 = \frac{1 * (2^8 - 1)}{2 - 1}](https://tex.z-dn.net/?f=S_8%20%3D%20%5Cfrac%7B1%20%2A%20%282%5E8%20-%201%29%7D%7B2%20-%201%7D)
![S_8 = \frac{1 * (256 - 1)}{1}](https://tex.z-dn.net/?f=S_8%20%3D%20%5Cfrac%7B1%20%2A%20%28256%20-%201%29%7D%7B1%7D)
![S_8 = \frac{1 * 255}{1}](https://tex.z-dn.net/?f=S_8%20%3D%20%5Cfrac%7B1%20%2A%20255%7D%7B1%7D)
![S_8 = 255](https://tex.z-dn.net/?f=S_8%20%3D%20255)
Solving (b): The entire board has 64 cells.
So:
![n = 64](https://tex.z-dn.net/?f=n%20%3D%2064)
Substitute 64 for n, 2 for r and 1 for a in ![S_n = \frac{a(r^n - 1)}{r - 1}](https://tex.z-dn.net/?f=S_n%20%3D%20%5Cfrac%7Ba%28r%5En%20-%201%29%7D%7Br%20-%201%7D)
![S_{64} = \frac{1 * (2^{64} - 1)}{2 -1}](https://tex.z-dn.net/?f=S_%7B64%7D%20%3D%20%5Cfrac%7B1%20%2A%20%282%5E%7B64%7D%20-%201%29%7D%7B2%20-1%7D)
![S_{64} = \frac{(2^{64} - 1)}{1}](https://tex.z-dn.net/?f=S_%7B64%7D%20%3D%20%5Cfrac%7B%282%5E%7B64%7D%20-%201%29%7D%7B1%7D)
![S_{64} = \frac{(18,446,744,073,709,551,616 - 1)}{1}](https://tex.z-dn.net/?f=S_%7B64%7D%20%3D%20%5Cfrac%7B%2818%2C446%2C744%2C073%2C709%2C551%2C616%20-%201%29%7D%7B1%7D)
![S_{64} = \frac{18,446,744,073,709,551,615}{1}](https://tex.z-dn.net/?f=S_%7B64%7D%20%3D%20%5Cfrac%7B18%2C446%2C744%2C073%2C709%2C551%2C615%7D%7B1%7D)
![S_{64} = 18,446,744,073,709,551,615](https://tex.z-dn.net/?f=S_%7B64%7D%20%3D%2018%2C446%2C744%2C073%2C709%2C551%2C615)