Find the sum of the geometric series. 20-5+5/4-5/16

The sum of the geometric series is 15.9375

Step-by-step explanation:

* Lets revise the geometric series

- There is a constant ratio between each two consecutive numbers

Ex:

5  ,  10  ,  20  ,  40  ,  80  ,  ………………………. (×2)

5000  ,  1000  ,  200  ,  40  ,  …………………………(÷5)

* General term (nth term) of a Geometric series:

- U1 = a  ,  U2  = ar  ,  U3  = ar²  ,  U4 = ar³  ,  U5 = ar^4

- Un = ar^n-1, where a is the first term , r is the constant ratio

 between each two consecutive terms  and n is the position

 of the number in the series

* The sum of first n terms of a Geometric series is calculated from

  the formula Sn = a(1 - r^n)/1 - r

* Lets solve the problem

∵ The geometric series is 20 , -5 , 5/4 , -5/16

∴ n = 4

∴ a = 20

∵ r = second term /first term

∴ r = -5/20 = -1/4

∵ Sn = a(1 - r^n)/1 - r

∴ S4 = 20(1 - (-1/4)^4)/(1 - (-1/4))

# Note: we neglect the -ve sign with the even power

∴ S4 = 20(1 - (1/256))/1 + 1/4 = 20(255/256)/(5/4)

∴ S4 = (1275/64)/(5/4) = 15.9375

* The sum of the geometric series is 15.9375

1 5/12

Step-by-step explanation:

2 1/3 = 7/4

3 3/4 = 15 / 4

The amount they eat = The amount they picked - The amount they remained

The amount they eat = 15 / 4 - 7/3

=45-28 / 12

=17/12

= 1 5/12