Question

Find S for the given geometric series. Round answers to the nearest hundredth, if necessary. a1 = –12, a5 = –7,500, r = 5 Question 4 options: –9,372 –6,252 –1,872 –18,780

Answer 1

S = -9,372 ⇒ 1st answer

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 = ar4

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

each two consecutive terms

- The sum of first n terms of a geometric series is calculate from

`S_(n)=(a(1-r^(n)))/(1-r)`

* Lets solve the problem

∵ The series is geometric

∵ a1 = -12

∴ a = -12

∵ a5 = -7500

∵ a5 = ar^4

∴ -7500 = -12(r^4) ⇒ divide both sides by -12

∴ 625 = r^4 take root four to both sides

∴ r = ± 5

∵ r = 5 ⇒ given

∵
`Sn=(a(1-r^(n)))/(1-r)`

∵ n = 5

∴
`S_(5)=(-12[1-(5)^(5)])/(1-5)=(-12[1-3125])/(-4)=3[-3124]=-9372`

* S = -9,372