Question

Please help me, thank you!

Answer 1

Answer:

a. The first four terms are -4 , -4/3, -4/9 , -4/27

b. The series is converge

c. The series has sum to ∞ , the sum of the series is -6

Step-by-step explanation:

* Lets revise the geometric series

- 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 Progression:

- U1 = a  ,  U2  = ar  ,  U3  = ar2  ,  U4 = ar3  ,  U5 = ar4

- 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 sequence

* In the problem

∵ The Un = -4(1/3)^n-1

∴ a = -4

∴ r = 1/3

a) To find the first four numbers use n = 1, 2 , 3 , 4

∴ U1 = a = -4

∴ U2 = -4(1/3)^(2 - 1) = -4(1/3) = -4/3

∴ U3 = -4(1/3)^(3 - 1) = -4(1/3)^2 = -4(1/9) = -4/9

∴ U4 = -4(1/3)^(4 - 1) = -4(1/3)^3 = -4(1/27) = -4/27

* The first four terms are -4 , -4/3, -4/9 , -4/27

b) If IrI < 1  then the geometric series is converge and if IrI > 1

   then the geometric series is diverge

∵ r = 1/3

∴ The series is converge  

c. The convergent series has sum to ∞

- The rule is: S∞ = a/(1 - r)

∴ S∞ = -4/(1 - 1/3) = -4/(2/3) = -4 × 3/2 = -6

* The sum of the series is -6