1. Find the common ratio and 9th term of the sequence 6, -3, 3/2, -3/4, .....

2. Find the 7th term of the sequence: 2, 6, 18, 54, .....

3. Find the 11th term of the sequence 1, -1/2, 1/4, -1/8, .....

well, r = -3/6 = (3/2)/-3 = (-3/4)/(3/2) ...

6/2 = ?

(-1/2)/1 = ?

Oh, sequences! Time to put my jester hat on and answer your questions!

1. Ah, the common ratio. Let's see, we start with 6, then we divide by -2, divide by 2 again, divide by -2 once more, and so on. So, the common ratio is -2. As for the 9th term, be ready to dive into fractions! It would be -3/256. Quite the shrinking sequence!

2. Ah, are we multiplying by 3 each time? It's like watching a number go to the gym and pump some iron! So, let's check the calculations: 2, 2 * 3 = 6, 6 * 3 = 18, 18 * 3 = 54. So, it seems like we're multiplying by 3 each time! With that pattern, the 7th term would be 2 * 3 * 3 * 3 * 3 * 3 * 3, which equals... *drumroll* ... 1458! That's some serious multiplication workout!

3. Ah, another shrinking sequence! It seems like we're dividing by 2 each time. So, the pattern goes like this: 1, 1/2, 1/4, 1/8... The 11th term would be 1/(2^11), which is 1/2048. It's like the numbers are playing hide-and-seek, but we've found them!

Remember, math can be serious, but sometimes it's good to have a laugh along the way! If you need anything else, I'm here to woo you with math jokes!

1. To find the common ratio, we divide each term by its preceding term. Let's calculate the common ratio:

-3 / 6 = -1/2
(3/2) / (-3) = -1/2
(-3/4) / (3/2) = -1/2

So, the common ratio of the sequence is -1/2.

To find the 9th term, we can use the formula for the nth term of a geometric sequence:

an = a1 * r^(n-1),

where a1 is the first term, r is the common ratio, and n is the term number.

Given that a1 = 6, r = -1/2, and n = 9, let's substitute the values into the formula:

a9 = 6 * (-1/2)^(9-1)
a9 = 6 * (-1/2)^8
a9 = 6 * (1/256)
a9 = 6/256
a9 = 3/128

So, the 9th term of the sequence is 3/128.

2. To find the 7th term of the sequence, we need to find the pattern. Looking at the sequence:

2, 6, 18, 54, ...

We can see that each term is obtained by multiplying the previous term by 3. Let's calculate the next term using this pattern:

54 * 3 = 162

So, the 7th term of the sequence is 162.

3. Similar to the previous problem, we can observe that each term is obtained by multiplying the preceding term by -1/2.

To find the 11th term, let's use the formula for the nth term of a geometric sequence:

an = a1 * r^(n-1)

Given a1 = 1, r = -1/2, and n = 11, we can substitute the values into the formula:

a11 = 1 * (-1/2)^(11-1)
a11 = 1 * (-1/2)^10
a11 = 1 * (1/1024)
a11 = 1/1024

So, the 11th term of the sequence is 1/1024.

1. To find the common ratio and 9th term of the sequence 6, -3, 3/2, -3/4, ..., we can examine the pattern of the sequence.

The sequence starts with 6, and then each subsequent term is obtained by multiplying the previous term by -1/2.

So, the common ratio, denoted by r, is -1/2.

To find the 9th term, denoted by t9, we can use the formula for the nth term of a geometric sequence:
tn = a * r^(n-1)

In this case, the first term a is 6, the common ratio r is -1/2, and the desired term n is 9.

t9 = 6 * (-1/2)^(9-1)
= 6 * (-1/2)^8
= 6 * (1/256)
= 6/256
= 3/128

Therefore, the common ratio is -1/2 and the 9th term of the sequence is 3/128.

2. To find the 7th term of the sequence 2, 6, 18, 54, ..., we can observe that each term is obtained by multiplying the preceding term by 3.

So, the common ratio, denoted by r, is 3.

To find the 7th term, denoted by t7, we can again use the formula for the nth term of a geometric sequence:
tn = a * r^(n-1)

In this case, the first term a is 2, the common ratio r is 3, and the desired term n is 7.

t7 = 2 * 3^(7-1)
= 2 * 3^6
= 2 * 729
= 1458

Therefore, the common ratio is 3 and the 7th term of the sequence is 1458.

3. To find the 11th term of the sequence 1, -1/2, 1/4, -1/8, ..., we can again observe the pattern.

Each term alternates between positive and negative, and is obtained by dividing the previous term by -2.

So, the common ratio, denoted by r, is -1/2.

To find the 11th term, denoted by t11, we can use the formula for the nth term of a geometric sequence:
tn = a * r^(n-1)

In this case, the first term a is 1, the common ratio r is -1/2, and the desired term n is 11.

t11 = 1 * (-1/2)^(11-1)
= 1 * (-1/2)^10
= 1 * (1/1024)
= 1/1024

Therefore, the common ratio is -1/2 and the 11th term of the sequence is 1/1024.