Question

For the geometric series given by 2+4+8+... , which of the following statements is FALSE?



Select one:

a. S200>S199

b. S200>a200

c. S1=a1

d. None of the other 3 statements here are False.

Answer 1

In a geometric sequence:

an = a1 ∙ rⁿ⁻¹

Sum of the first n terms of a geometric sequence:

Sn = a1 ∙ ( 1 − rⁿ ) / ( 1− r )

where

a1 = first term

r is the common ratio


In tis case

a1 = 2 , r = 2

so

Sn = a1 ∙ ( 1 − rⁿ ) / ( 1− r )


S199 = 2 ∙ ( 1 − 2¹⁹⁹ ) / ( 1− 2 )

S199 = 2 ∙ ( 1 − 2¹⁹⁹ ) / ( - 1 )

S199 = - 2 ∙ ( 1 − 2¹⁹⁹ ) = 1.6069380442589902755419620923412 ∙ 10⁶⁰


S200 = 2 ∙ ( 1 − 2²⁰⁰ ) / ( 1− 2 )

S200 = 2 ∙ ( 1 − 2²⁰⁰ ) / ( - 1 )

S200 = - 2 ∙ ( 1 − 2²⁰⁰ ) = 3.2138760885179805510839241846823 ∙ 10⁶⁰



S200 > S199

3.2138760885179805510839241846823 ∙ 10⁶⁰ > 1.6069380442589902755419620923412 ∙ 10⁶⁰

True


an = a ∙ rⁿ⁻¹

a200 = 2 ∙ 2¹⁹⁹ = 2²⁰⁰ = 1.6069380442589902755419620923412 ∙ 10⁶⁰


S200 > a200

3.2138760885179805510839241846823 ∙ 10⁶⁰ > 1.6069380442589902755419620923412 ∙ 10⁶⁰

True


S1 = a1

2 = 2

True


d. None of the other 3 statements here are False.