Question

Find the sum of the following infinite geometric series if it exists. (2/5) + (12/25) + (72/125) +...

	A.	5/12
	B.	Does not exist
	C.	5/6
	D.	6/5

Which of the following is not a geometric sequence? 

	A.	1, 2, 4, 8
	B.	3, 1, -1,-3
	C.	½, ¼, 1/8, 1/16
	D.	216, 72, 24, 8

 

Answer 1

(1).

The geometric series is (2/5) + (12/25) + (72/125) + · · · · · · · .

To calculate the sum of the infinite geometric series by using formula : S_∞ = a₁/(1 - r), - 1 < r < 1 ; where a₁ = first term and r = common ratio.

a₁ = 2/5 and r = a₂ / a₁ = (12/25) / (2/5) = 6/5 = 1.2.

Since r = 1.2 does not lie in the interval (- 1, 1), the sum of the infinite geometric series does not exist.

The option B is correct.

Answer 2

(2).

The sequence is 1, 2, 4, 8.

First find the ratios of consecutive terms.

2/1 = 4/2 = 8/4 = 2.

The ratios of consecutive terms are the same, so sequence is geometric sequence.

 

The sequence is 3, 1, - 1, - 3.

First find the ratios of consecutive terms.

1/3 ≠ - 1/1 ≠ (-3)/(-1).

The ratios of consecutive terms are not same, so sequence is not geometric sequence.

 

The sequence is 1/2, 1/4, 1/8, 1/16.

First find the ratios of consecutive terms.

(1/4) / (1/2) = (1/8) / (1/4) = (1/16) / (1/8) = 1/2.

The ratios of consecutive terms are the same, so sequence is geometric sequence.

 

The sequence is 216, 72, 24, 8.

First find the ratios of consecutive terms.

72 / 216 = 24 / 72 = 8 / 24 = 1/3.

The ratios of consecutive terms are the same, so sequence is geometric sequence.

 

The option B is correct.