<PRE><B>Question 25545</B>
Ok, this is one that has baffled not only myself but my parents and boyfriend as well. This is a problem that no one seems to be able to figure out. Do you think you could help me? 
GOOD PROBLEM YOUNG GIRL!NOW THAT YOU SAY THAT YOUR BOY FRIEND IS ALSO FOXED BY THE PROBLEM ,YOU CAN USE IT TO GOOD EFFECT TO PUT ACROSS A PROPOSAL TO HIM ...
YOU PROMISS HIM A 100,000..POUNDS EVERY DAY FOR 30 DAYS.LET HIM GIVE YOU IN RETURN JUST A PENNY ON THE FIRST DAY,2 PENNIES ON THE SECOND DAY ,4 PENNIES ON THE THIRD DAY,......ETC FOR 30 DAYS ..AS IN YOUR PROBLEM...BASED ON YOUR FEED BACK , I PRESUME HE WILL LAP ON THE PROPOSAL AND I CAN ASSURE YOU THAT YOU WONT REGRET THE PROPOSAL!!!!
OK NOW LET US GET BACK TO YOUR PROBLEM AS WELL AS MY SUGGESTION ,WHICH IS JUST A PART OF IT .I AM GIVING BELOW THE MONEY TO BE PAID ON THIS BASIS FOR 30 AND 64 DAYS ,ROUNDED OFF TO POUNDS OR MILLION POUNDS AT THE LATER DAYS.

DAY.....MONEY TO BE PAID........MONEY TO BE PAID BY YOUR FRIEND TO YOU IN 		
............TO YOUR FRIEND.....	CENTS...POUNDS...MILLION POUNDS
................POUNDS				
1	100000		1	0	0
2	100000		2	0	0
3	100000		4	0	0
4	100000		8	0	0
5	100000		16	0	0
6	100000		32	0	0
7	100000		64	1	0
8	100000		128	1	0
9	100000		256	3	0
10	100000		512	5	0
11	100000		1024	10	0
12	100000		2048	20	0
13	100000		4096	41	0
14	100000		8192	82	0
15	100000		16384	164	0
16	100000		32768	328	0
17	100000		65536	655	0
18	100000		131072	1311	0
19	100000		262144	2621	0
20	100000		524288	5243	0
21	100000		1048576	10486	0
22	100000		2097152	20972	0
23	100000		4194304	41943	0
24	100000		8388608	83886	0
25	100000		16777216	167772	0
26	100000		33554432	335544	0
27	100000		67108864	671089	1
28	100000		134217728	1342177	1
29	100000		268435456	2684355	3
30	100000		536870912	5368709	5	
31	100000		1073741824	10737418	11	
32	100000		2147483648	21474836	21	
33	100000		4294967296	42949673	43	
34	100000		8589934592	85899346	86	
35	100000		17179869184	171798692	172	
36	100000		34359738368	343597384	344	
37	100000		68719476736	687194767	687	
38	100000		1.37439E+11	1374389535	1374	
39	100000		2.74878E+11	2748779069	2749	
40	100000		5.49756E+11	5497558139	5498	
41	100000		1.09951E+12	10995116278	10995	
42	100000		2.19902E+12	21990232556	21990	
43	100000		4.39805E+12	43980465111	43980	
44	100000		8.79609E+12	87960930222	87961	
45	100000		1.75922E+13	175921860444	175922	
46	100000		3.51844E+13	351843720888	351844
47	100000		7.03687E+13	703687441777	703687
48	100000		1.40737E+14	1407374883553	1407375
49	100000		2.81475E+14	2814749767107	2814750
50	100000		5.6295E+14	5629499534213	5629500
51	100000		1.1259E+15	11258999068426	11258999
52	100000		2.2518E+15	22517998136853	22517998
53	100000		4.5036E+15	45035996273705	45035996
54	100000		9.0072E+15	90071992547410	90071993
55	100000		1.80144E+16	180143985094820	180143985
56	100000		3.60288E+16	360287970189640	360287970
57	100000		7.20576E+16	720575940379279	720575940
58	100000		1.44115E+17	1441151880758560	1441151881
59	100000		2.8823E+17	2882303761517120	2882303762
60	100000		5.76461E+17	5764607523034230	5764607523
61	100000		1.15292E+18	11529215046068500	11529215046
62	100000		2.30584E+18	23058430092136900	23058430092	
63	100000		4.61169E+18	46116860184273900	46116860184	
64	100000		9.22337E+18	92233720368547800	92233720369	
						    
30 DAY..3000000.........1073741823......10737418.....................11	
TOTAL						
						
64 DAY..6400000.........1.84467E+19.....184467440737095000......184467440737	
TOTAL
 
YOUR GAIN IN 30 DAYS = 8 MILLION POUNDS..GOT IT BUT ONLY BE CAREFULL THAT YOUR FRIEND WONT RUN OUT OF YOU ON THE 25 TH. DAY.!!!

YOUR GAIN IN 30 DAYS  = 184467440731 MILLION POUNDS

NOW COMING TO THE MATHS PART OF THIS ,THIS SEQUENCE WHERE EACH NUMBER BEARS A CONSTANT RATIO TO ITS PREDECESSOR IS CALLED GEOMETRIC PROGRESSION..HERE YOU FIND EACH NUMBER IS OBTAINED FROM THE PREVIOUS ONE BY MULTIPLYING WITH 2 , CALLED COMMON RATIO.
THE LAST NUMBER ON NTH. DAY AND SUM OF SUCH SERIES OF NUMBERS UPTO THE N TH.DAY IS GIVEN BY THE FOLLOWING FORMULAE...
N TH. NUMBER = FIRST NUMBER *(C0MM0N RATIO )^(N-1)=A*(R)^(N-1)=(2)^(N-1) IN THIS CASE.
SUM UP TO N TH.NUMBER =A*{((R)^N - 1))/(R-1)}={(2)^N-1}IN THIS CASE</PRE>