Show that each sequence is geometric. Find the common ratio and write out the first four terms. (Can you explain how to do it too?)

> {(5/2)^n)
> {(3n/9)}
> {3^2n}

a.

r= (5/2)^(n+1) / (5/2)^n = 5/2

b. do the same thing on the last two, compare the ratio of the n+1 term t the n term. If it is constant, it is geometric.

To determine whether a sequence is geometric, we need to check if there is a common ratio between consecutive terms.

Let's go through each sequence one by one:

1) {(5/2)^n}
To check if this sequence is geometric, we need to see if there is a common ratio between consecutive terms. To find the common ratio, we can divide any term by its previous term. In this case, let's divide the second term by the first term: ((5/2)^2) / ((5/2)^1) = (25/4)/(5/2) = 25/4 * 2/5 = 25/10 = 5/2.

Since the ratio between consecutive terms is constant at 5/2, we can conclude that the sequence {(5/2)^n} is geometric.

The common ratio is 5/2, and the first four terms are:
n=1: (5/2)^1 = 5/2
n=2: (5/2)^2 = 25/4
n=3: (5/2)^3 = 125/8
n=4: (5/2)^4 = 625/16

2) {(3n/9)}
Similarly, let's check if this sequence is geometric by finding the common ratio. Divide the second term by the first term: ((3(2))/9) / ((3(1))/9) = 6/9 = 2/3.

Since the ratio between consecutive terms is constant at 2/3, we can conclude that the sequence {(3n/9)} is geometric.

The common ratio is 2/3, and the first four terms are:
n=1: (3(1))/9 = 1/3
n=2: (3(2))/9 = 2/3
n=3: (3(3))/9 = 1
n=4: (3(4))/9 = 4/3

3) {3^(2n)}
To determine if this sequence is geometric, we once again find the common ratio by dividing the second term by the first term: (3^(2(2))) / (3^(2(1))) = (3^4) / (3^2) = 81/9 = 9.

Since the ratio between consecutive terms is constant at 9, we can conclude that the sequence {3^(2n)} is geometric.

The common ratio is 9, and the first four terms are:
n=1: 3^(2(1)) = 3^2 = 9
n=2: 3^(2(2)) = 3^4 = 81
n=3: 3^(2(3)) = 3^6 = 729
n=4: 3^(2(4)) = 3^8 = 6561

By applying the concept of finding the common ratio and observing its constancy, we can determine whether a sequence is geometric and find the common ratio and the first few terms.