SOLUTION: As I was going to St. Ives, I met a man with 7 wives, every wife had 7 sacks, and every sack had 7 cats. Every cat had 7 kits. Kits, cats, sacks, wives, how many were going to St.

Algebra ->  Permutations -> SOLUTION: As I was going to St. Ives, I met a man with 7 wives, every wife had 7 sacks, and every sack had 7 cats. Every cat had 7 kits. Kits, cats, sacks, wives, how many were going to St.       Log On


   

Question 929247: As I was going to St. Ives, I met a man with 7 wives, every wife had 7 sacks, and every sack had 7 cats. Every cat had 7 kits. Kits, cats, sacks, wives, how many were going to St. Ives? A.) use a geometric series to model as a sum the number of items (kits,cats,sacks,wives,man) that the narrator met. B.) evaluate this geometric series in 2 ways: directly by adding the terms, and also by using the summation formula that was developed.
Answer by KMST(5328) About Me  (Show Source):
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A)
The first term is t%5B1%5D=1 for 1 man.
The second term is t%5B2%5D=1%2A7=7 for the 7 wives per man for that 1 man.
The third term is t%5B3%5D=t%5B2%5D%2A7=7%2A7=49 for the 7 sacks per wife times 7 wives.
We see that we have a geometric sequence, with common ratio r=7 , and each term will be according to t%5Bn%5D=t%5B1%5D%2Ar%5E%28n-1%29=1%2A7%5E%28n-1%29=7%5E%28n-1%29
The fourth term is t%5B4%5D=7%5E3=343 representing the total number of cats.
The fifth term is t%5B5%5D=7%5E4=2401 representing the total number of cats.
B) The sum the number of items (kits,cats,sacks,wives,man) that the narrator met is
1%2B7%2B49%2B343%2B2401=2801 (directly by adding the terms).
Since the sum of a geometric series can be calculates as
S%5Bn%5D=t%5B1%5D%2A%28%28r%5En-1%29%2F%28r-1%29%29 ,
in this case the sum is S%5B5%5D=1%2A%28%287%5E5-1%29%2F%287-1%29%29=%2816807-1%29%2F6=16806%2F6=2801