Please give the answers and solutions for each.

1.If the second term is 2 and the seventh term of a geometric sequence is 64, find the 12th term.

2. Which term if the geometric sequence 18,54,162,486,... is 3,188,646?

3. Determine the geometric mean of 8 and 288

4.Find x if the geometric mean of 6 and x is 3sqrt2

5. Find two geometric sequences where the first term is 1 and the fifth term is 256

6. Insert the indicated number of positive geometric means between each pair of numbers.
a. 2 and 31.25 [2]
b. 17 and 272 [3]
c.1/5 and 1/15625 [4]
*What does this mean [number] ?(the one in my abc's questions)

7.What is the sum of the first 10 terms of the geometric series 6+18+54+162+...?

8. What is the sum of each infinite geometric series if it exists ?
a.3-2+4/3-8/9+...
b. -1-0.1-0.01-0.001-...
c. sqrt2+2+sqrt8+4+...

9. A rubber ball rebounds 3/5 of the height from which it falls. If it is initially dropped from a height of 30 meters, what total vertical distance does it travel before coming to rest ?

*CHALLENGE
1. The Fibonacci sequence is defined recursively by a sub 1 = a sub 2 = 1 and a sub n+2 = a sub n+1 + a sub n for n > 1 . The first 5 terms of the said sequence is 1,1,2,3,5. what is the sum of the digits of the LCM of the first 10 terms of the Fibonacci sequence ?

2. In the triangular array of inters at the right, what the sum of the intergers in the 100th row ?
1
3 5
7 9 11
13 15 17 19
21 23 25 27 29
31 33 35 37 39 41

3. The sum of the first 4 terms of an arithmetic sequence a sub 1, a sub2 , a sub 3, a sub 4 is 2012 and the said 4 terms also form a geomeric sequence . Find a sub 4.

4. the 7th term of a geometric sequence is 20 and the 13th term is 12. Find the 10th term.

5. how many terms of the sequence -7,-4,-1,2,5,8,... must be taken to get a sum of 143?

6. the 5oth term of an arithmetic sequence exceeds the 10th term by 240. if the last digit of the 50th term is 8, what is the last digit of the 2012th term ?

7. the arithmetic mean of two positive integers is 1/2 more than their geomtric mean. if one of the integers is 1, what is the other integer ?

8.Find the sum of the first 2012 terms of the sequence 1+3-5+7+9-11+13+15-17+...

9.the sum of the arithmetic mean and the geometric mean of 2 positive numbers is 2012. what is the sum of the positive square roots of the 2 numbers ?

10. wat is the sum of 3 distinct positive integers whose geometric mean is 7 ?

1. To find the 12th term of a geometric sequence, we need to know either the common ratio or another term of the sequence. Given that the second term is 2 and the seventh term is 64, we can find the common ratio by dividing the seventh term by the second term:

Common ratio = 64 / 2 = 32

Now, to find the 12th term, we can use the formula for the nth term of a geometric sequence:

nth term = a * r^(n-1)

where a is the first term and r is the common ratio. Since we have the second term (a = 2) and the common ratio (r = 32), we can substitute these values into the formula:

12th term = 2 * 32^(12-1)
= 2 * 32^11

2. To find the term in a geometric sequence, we need the first term and the common ratio. Given the geometric sequence 18, 54, 162, 486, and the term 3,188,646, we can try to find the common ratio by dividing any term by its previous term:

Common ratio = 54 / 18 = 3

Now, to find the term number, we can use the formula for the nth term:

Term number = log(3,188,646 / 18) / log(3)

3. The geometric mean of two numbers is the square root of their product. To find the geometric mean of 8 and 288, we can simply calculate the square root of their product:

Geometric mean = sqrt(8 * 288)

4. The geometric mean of two numbers is the square root of their product. Given that the geometric mean of 6 and x is 3√2, we can set up an equation:

sqrt(6 * x) = 3√2

5. To find two geometric sequences with the first term as 1 and the fifth term as 256, we need the common ratio. Let's call the common ratio of the first sequence "r" and the common ratio of the second sequence "s". We can set up two equations:

1 * r^4 = 256 (First sequence)
1 * s^4 = 256 (Second sequence)

Solving these equations will give us the common ratios for the two sequences.

6. When the prompt mentions inserting a number of positive geometric means between two numbers, it means finding the intermediate terms in a geometric sequence. The number within the square brackets (e.g., [2], [3], [4]) represents the number of geometric means or intermediate terms that need to be inserted between the given pairs of numbers.

For example, in the case of "Insert 2 positive geometric means between 2 and 31.25," we need to find two numbers that fit into a geometric sequence between 2 and 31.25.

To find the geometric means, we can use the formula:

Geometric mean = sqrt(a * b)

where a and b are the given numbers.

7. To find the sum of the first 10 terms of a geometric series, we need to know the first term, the common ratio, and the number of terms in the series. Given the geometric series 6 + 18 + 54 + 162 + ..., we can find the first term (a) and the common ratio (r) by taking the ratio of any term to its previous term:

Common ratio = 18 / 6 = 3

Now, we can use the formula for the sum of a geometric series:

Sum of first n terms = a * (r^n - 1) / (r - 1)

where a is the first term, r is the common ratio, and n is the number of terms. Substituting the values into the formula:

Sum of first 10 terms = 6 * (3^10 - 1) / (3 - 1)

8. For infinite geometric series, the sum exists only if the absolute value of the common ratio is less than 1. Let's examine each given series:

a. 3 - 2 + 4/3 - 8/9 + ...
b. -1 - 0.1 - 0.01 - 0.001 - ...
c. sqrt2 + 2 + sqrt8 + 4 + ...

To find the sum, we can use the formula for the sum of an infinite geometric series:

Sum = a / (1 - r)

where a is the first term and r is the common ratio.

9. When a rubber ball rebounds, it returns to a certain fraction of the height from which it fell. In this case, it rebounds 3/5 of the height. To find the total vertical distance the ball travels before coming to rest, we need to consider both the heights it falls and rebounds from.

Given that the initial height is 30 meters, we can find the total distance by summing up the heights the ball falls and rebounds from:

Total distance = 30 + 30 * (3/5) + 30 * (3/5)^2 + 30 * (3/5)^3 + ...

Since the ball rebounds repeatedly, we have an infinite geometric series. The common ratio is 3/5, and the first term is 30. To find the sum, we can use the formula:

Sum = a / (1 - r)

where a is the first term and r is the common ratio. In this case, a = 30 and r = 3/5.