Andy C. answered • 08/02/18

Tutor

4.9
(27)
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