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
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