Write the first five terms of the geometric sequence in which a_1=64 and the common ratio is 5/4 .
64,80,100,125,625/4
64,36,27,81/4,243/16
64,16,4,1,1/4
64,16,1/4,5/16,25/64
The correct answer is the last option: 64, 16, 1/4, 5/16, 25/64
show the steps
To find the terms of the geometric sequence, we can use the formula:
a_n = a_1 * r^(n-1)
where a_n is the nth term of the sequence, a_1 is the first term, r is the common ratio, and n is the position of the term.
Given that a_1 = 64 and the common ratio is 5/4, the sequence can be written as:
a_n = 64 * (5/4)^(n-1)
To find the first five terms, we substitute n = 1, 2, 3, 4, and 5 into the formula.
For n = 1:
a_1 = 64 * (5/4)^(1-1)
= 64 * (5/4)^0
= 64 * 1
= 64
For n = 2:
a_2 = 64 * (5/4)^(2-1)
= 64 * (5/4)^1
= 64 * 5/4
= 320/4
= 80
For n = 3:
a_3 = 64 * (5/4)^(3-1)
= 64 * (5/4)^2
= 64 * 25/16
= 1600/16
= 100
For n = 4:
a_4 = 64 * (5/4)^(4-1)
= 64 * (5/4)^3
= 64 * 125/64
= 8000/64
= 125
For n = 5:
a_5 = 64 * (5/4)^(5-1)
= 64 * (5/4)^4
= 64 * 625/256
= 40000/256
= 625/4
The first five terms of the geometric sequence are: 64, 80, 100, 125, and 625/4.
Explain how to tell if a geometric series is convergent or divergent. Include an example of a convergent series and an example of a divergent series in your explanation.
To determine if a geometric series is convergent or divergent, we need to look at the common ratio (r) of the sequence.
For a geometric series to be convergent, the absolute value of the common ratio (|r|) must be less than 1. In other words, the terms of the sequence must approach zero as n approaches infinity. When this condition is met, we can use the formula for the sum of a geometric series to determine the convergence.
The formula for the sum of a convergent geometric series is given by:
S = a_1 / (1 - r)
where S is the sum of the series, a_1 is the first term, and r is the common ratio.
Example of a convergent geometric series:
Consider the series: 3, 3/2, 3/4, 3/8, ...
In this series, the first term (a_1) is 3 and the common ratio (r) is 1/2.
|1/2| = 1/2 < 1
Since the absolute value of the common ratio is less than 1, the series is convergent. We can find the sum of this series using the formula:
S = 3 / (1 - 1/2) = 3 / (1/2) = 3 * 2/1 = 6
So, the sum of the series is 6.
For a geometric series to be divergent, the absolute value of the common ratio (|r|) must be equal to or greater than 1. In this case, the terms of the sequence do not approach zero, and the series does not have a finite sum.
Example of a divergent geometric series:
Consider the series: 5, 10, 20, 40, ...
In this series, the first term (a_1) is 5 and the common ratio (r) is 2.
|2| = 2 > 1
Since the absolute value of the common ratio is greater than 1, the series is divergent. The terms in the series continue to grow larger without bound, and there is no finite sum.
In conclusion, we can determine if a geometric series is convergent or divergent by examining the absolute value of the common ratio. If |r| < 1, the series is convergent and has a finite sum. If |r| ≥ 1, the series is divergent and does not have a finite sum.