Need help with a math sequence problem?

Question

If you have a checkerboard and you place one penny on the first square and two pennies on the second square and then four pennies on the third square...continue until all 64 square are covered with pennies.


How much money expressed in dollars would you have to put on the 32nd square? Please show how to arrive at answer.


How much money expressed in dollars would you receive in total if the checkerboard only had 32 squares? Please show how to arrive at answer.


Calculate the amount of money necessary to fill the whole checkerboard (64 squares). How much money expressed in dollars would their be?Please show work.

Answer 1

2^1 + 2^2 + 2^3 + ... + 2^64
(two cents doubled each time, added together)

Series where 2^n
for n = 1 to 64 for 64 squares
(or for n = 1 to 32 for 32 squares)

Note 1: This is added in "cents" - - to get
"dollars" remember to move the decimal
back to the left two places. So 200 cents = 2.00 dollars

Note 2: I tried to list these on an Excel sheet
but the numbers start "rounding up" and you
have to add them by hand. Please see below.

2
4
8
16
32
64
128
256
512
1024
2048
4096
8192
16384
32768
65536
131072
262144
524288
1048576
2097152
4194304
8388608
16777216
33554432
67108864
134217728
268435456
536870912
1073741824
2147483648
4294967296

Rows 1 through 32 work
(should add up to about
8589934590 if sum is correct)

But rows 33 through 64 have to be "split up" manually
or else the Excel program rounds off to the nearest 0:

8589934592
17179869184
34359738368
68719476736
137438953472
274877906944
549755813888
1099511627776
2199023255552
4398046511104
8796093022208
17592186044416
35184372088832
70368744177664
140737488355328
281474976710656
562949953421312

Below this point, Excel started rounding off to 0,
so I put the difference in a second column
and started adding these manually. Otherwise,
you may introduce rounding errors in your data:

1125899906842620+ 4
2251799813685240+ 8
4503599627370480+ 16
9007199254740960+ 32

18014398509481900 84
36028797018963800 168
72057594037927600 336
144115188075855000 872
288230376151710000 1744
576460752303420000 3488
1152921504606840000 6976
2305843009213680000 13952
4611686018427360000 27904
9223372036854720000 55808
18446744073709400000 111616

So to get the actual numbers for each row across,
you would have to check both numbers (to make sure there are no rounding errors) and add them together.

And to the get the total sum, you would have
to add all the numbers going down.

I don't know if this is correct or rounded by Excel.
But at least you can check your work against this list
and see if you get the same numbers or totals.

Note: Check by adding manually or with another system.
When I split up the numbers starting with row 59 - 64
I got different answers:

562949953420000 + 1312
1125899906840000 + 2624
2251799813680000 5248
4503599627360000 10496
9007199254720000 20992
18014398509440000 41984
36028797018880000 83968
72057594037760000 167936
144115188075520000 335872
288230376151040000 671744
576460752302080000 1343488
1152921504604160000 2686976
2305843009208320000 5373952
4611686018416640000 10747904
9223372036833280000 21495808
18446744073666600000 42991616