Respuesta :
Answer:
A. Yes.
B. Yes.
C. No.
Step-by-step explanation:
A. Yes. The sum of the series, [tex]5 + 25x + 125x^{2} + 625x^{3} + 3125x^{4} + ...........[/tex] is the sum of a geometric series.
The first term of the series [tex]a_n[/tex] = 5.
The common ration or the ratio between successive terms (r) = [tex]\frac{25x}{5} = 5x[/tex] (Answer)
B. Yes. The sum of the series, [tex]5x^{7} + 5x^{8} + 5x^{9} + 5x^{10} + ............[/tex] is also the sum of a geometric series.
The first term of the series [tex]a_n = 5x^{7}[/tex].
The common ration or the ratio between successive terms (r) = [tex]\frac{5x^{8} }{5x^{7} } = x[/tex] (Answer)
C. No. The sum of the series, [tex]5 + 5x + 5x^{2} + 5x^{4} + 5x^{8} + ...........[/tex] is not the sum of a geometric series.
The first term of the series [tex]a_n = 5[/tex].
(Answer)
Geometric series have the ratio of any two consecutive terms is the same. The first two series are geometric series, while the last one is not a geometric series.
What is a geometric series?
A geometric series is a series in which the ratio of any two consecutive terms is the same.
The given series will be a geometric series if the ratio of two consecutive terms of the series is the same.
A.) [tex]5+25x+125x^2+625x^3+3125x^4+...[/tex]
As we can see that the ratio of any two consecutive terms of the series is 5x, therefore, the series is a geometric series.
The first term of the series is 5.
B.) [tex]5x^7+5x^8+5x^9+5x^{10}+....[/tex]
As we can see that the ratio of any two consecutive terms of the series is x, therefore, the series is a geometric series.
The first term of the series is 5.
C.) [tex]5+5x+5x^2+5x^4+5x^8+....[/tex]
As we can see that the ratio of any two consecutive terms of the series is not the same, therefore, the series is not a geometric series.
The first term of the series is na.
Learn more about Geometric Series:
https://brainly.com/question/1504226
