Question

For each of the following, decide if the given series is a geometric series. A. 5+25x+125x2+625x3+3125x4+⋯ : Is this a geometric series? A. Yes B. No If this is a geometric series, enter the first term: na and the ratio between successive terms: na (Enter na for the first term and ratio if this is not a geometric series.) B. 5x7+5x8+5x9+5x10+⋯ : Is this a geometric series? A. Yes B. No If this is a geometric series, enter the first term: na and the ratio between successive terms: na (Enter na for the first term and ratio if this is not a geometric series.) C. 5+5x+5x2+5x4+5x8+⋯ : Is this a geometric series? A. Yes B. No If this is a geometric series, enter the first term: na and the ratio between successive terms: na (Enter na for the first term and ratio if this is not a geometric series.)

Answer 1

Answer:

A. Yes.

B. Yes.

C. No.

Step-by-step explanation:

A. Yes. The sum of the series, $5 + 25x + 125x^{2} + 625x^{3} + 3125x^{4} + ...........$ is the sum of a geometric series.

The first term of the series $a_n$ = 5.

The common ration or the ratio between successive terms (r) = $\frac{25x}{5} = 5x$ (Answer)

B. Yes. The sum of the series, $5x^{7} + 5x^{8} + 5x^{9} + 5x^{10} + ............$ is also the sum of a geometric series.

The first term of the series $a_n = 5x^{7}$.

The common ration or the ratio between successive terms (r) = $\frac{5x^{8} }{5x^{7} } = x$ (Answer)

C. No. The sum of the series, $5 + 5x + 5x^{2} + 5x^{4} + 5x^{8} + ...........$ is not the sum of a geometric series.

The first term of the series $a_n = 5$.

(Answer)