Answer:
255
4,294,967,295
18,446,744,073,709,551,615
Step-by-step explanation:
Geometric Sequence
Each term of a geometric sequence is obtained by multiplying (or dividing) the previous term by a constant number called common ratio (r). The recursive formula for each term n is
[tex]a_n=a_{n-1}\cdot r[/tex]
And the general formula is
[tex]a_n=a_1\cdot r^{n-1}[/tex]
for n>=1
We place 1 penny on the first cell (square) (row 1, column A). That is the value of a1=1. Now each new cell will have double pennies as much the previous cell. The common ratio is r=2. Thus, the following general formula gives us the number of pennies on the cell n
[tex]a_n= 2^{n-1}[/tex]
This gives us the sequence 1,2,4,8,16,32,...,[tex]2^{63}[/tex]
To get the sum of n terms, we notice the sum forms the sequence 1, 3, 7, 15, 31, 63,... which we can transform as
2-1, 4-1, 8-1, 16-1, 32-1, 64-1,...
Or equivalently
[tex]2^1-1,2^2-1,2^3-1,2^4-1,2^5-1,....2^{n}-1[/tex]
The formula for the sum of n terms is, then
[tex]S_n=2^{n}-1[/tex]
Now we are ready to answer the questions
- How many pennies in Row 1?
Row 1 has n=8 cells, thus
[tex]S_8=2^{8}-1=255[/tex]
- How many pennies in Row 1-4?
The first 4 rows contain 4*8=32 cells, thus
[tex]S_{32}=2^{32}-1=4,294,967,295[/tex]
- How many pennies on the entire chessboard?
Here n=64
[tex]S_{64}=2^{64}-1=18,446,744,073,709,551,615[/tex]
