
Andy C. answered 08/02/18
Tutor
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Math/Physics Tutor
part 1:
Sum of finite geometric series has partial sum:
Sn = [1 - r^(n+1) ]/ (1-r) where r is the common ratio
Since 0 < r < 1,
0 < r^2 < r < 1
0 < r^3 < r^2 < r < 1
....
0 < r^(n) < r^(n-1) < r < 1
Multiplying everything by 0<r<1,
0 < r^(n+1) < r^n < ... <r^2 < r < 1
So the statement hold for any positive integer 0<N<infinity
0 < r^(n) < 1
which shows the common ratio converges to zero. The sum converges as a result
part 2:
The sum IN PENNIES is 1 + 2 + 3 + 4 + .... + N which is the sum of the first N integers,where N is the # of weeks.
It can also be shown by induction that the sum of these first N integers is
N(N+1)/2
1 year is 52 weeks, so there will be
52*53/2 = 26*53 = 1378
$13.78