PLEASE HELP7.05a1. Find the first six terms of the sequence.a1 = -6, an = 4 • an-1 A) 0, 4, -24, -20, -16, -12 B) -24, -96, -384, -1536, -6144, -24,576 C) -6, -24, -20, -16, -12, -8 D) -6, -24, -96, -384, -1536, -61442. Find an equation for the nth term of the arithmetic sequence.-15, -6, 3, 12, ... A) an = -15 + 9(n + 1) B) an = -15 x 9(n - 1) C) an = -15 + 9(n + 2) D) an = -15 + 9(n - 1)3. Find an equation for the nth term of the arithmetic sequence.a14 = -33, a15 = 9 A) an = -579 + 42(n + 1) B) an = -579 + 42(n - 1) C) an = -579 - 42(n + 1) D) an = -579 - 42(n - 1)4. Determine whether the sequence converges or diverges. If it converges, give the limit. 48, 8, four divided by three , two divided by nine , ... A) Converges; two hundred and eighty eight divided by five B) Converges; 0 C) Diverges D) Converges; -124325. Find an equation for the nth term of the sequence.-3, -12, -48, -192, ... A) an = 4 • -3n + 1 B) an = -3 • 4n - 1 C) an = -3 • 4n D) an = 4 • -3n6. Find an equation for the nth term of a geometric sequence where the second and fifth terms are -21 and 567, respectively. A) an = 7 • (-3)n + 1 B) an = 7 • 3n - 1 C) an = 7 • (-3)n - 1 D) an = 7 • 3n7. Write the sum using summation notation, assuming the suggested pattern continues.5 - 15 + 45 - 135 + ... A) summation of five times three to the power of the quantity n plus one from n equals zero to infinity B) summation of five times negative three to the power of n from n equals zero to infinity C) summation of five times three to the power of n from n equals zero to infinity D) summation of five times negative three to the power of the quantity n plus one from n equals zero to infinity8. Write the sum using summation notation, assuming the suggested pattern continues.-9 - 3 + 3 + 9 + ... + 81 A) summation of the quantity negative nine plus six n from n equals zero to fifteen B) summation of negative fifty four times n from n equals zero to fifteen C) summation of negative fifty four times n from n equals zero to infinity D) summation of the quantity negative nine plus six n from n equals zero to infinity9. Write the sum using summation notation, assuming the suggested pattern continues.64 + 81 + 100 + 121 + ... + n2 + ... A) summation of n squared from n equals eight to infinity B) summation of n minus one squared from n equals eight to infinity C) summation of n squared from n equals nine to infinity D) summation of n plus one squared from n equals eight to infinity10. Find the sum of the arithmetic sequence.17, 19, 21, 23, ..., 35 A) 260 B) 179 C) 37 D) 16011. Find the sum of the geometric sequence. 1, one divided by two, one divided by four, one divided by eight, one divided by sixteen A) one divided by twelve B) 93 C) negative one divided by forty eight D) thirty one divided by sixteen12. An auditorium has 30 rows with 10 seats in the first row, 12 in the second row, 14 in the third row, and so forth. How many seats are in the auditorium? A) 1170 B) 735 C) 1230 D) 60013. Use mathematical induction to prove the statement is true for all positive integers n.The integer n3 + 2n is divisible by 3 for every positive integer n.14. A certain species of tree grows an average of 3.8 cm per week. Write an equation for the sequence that represents the weekly height of this tree in centimeters if the measurements begin when the tree is 5 meters tall.

These are 14 questions and 14 answers.

Since this exceeds the limit and I had to delete the last questions and I copied all the answers to a file that is attache. See the attachment with all the answers.

1) Question 1. Find the first six terms of the sequence: a1 = -6, an = 4 • an-1

Answer: option D) -6, -24, -96, -384, -1536, -6144

Explanation:

A(1) = - 6

A(n) = 4 * A(n-1)

n                 A(n)

1                 - 6

2                  4 * (-6) = - 24

3                  4 * (-24) = - 96

4                  4 * (-96) = - 384

5                  4 * (-384) = - 1536

6                  4 * ( -1536) = -6144

So, the first six terms are: -6, - 24, - 96, - 384, - 1536, - 6144.

2) Question 2: Find an equation for the nth term of the arithmetic sequence.
-15, -6, 3, 12, ...

Answer: option D) - 15 + 9(n - 1)

Explanation:

1. find the difference between the consecutive terms:

-6 - (-15) = -6 + 15 = 9
3 - (-6) = 3 + 6 = 9
12 - 3 = 9

So, the difference is 9, and you can find any term adding 9 to the previous.

2. Since the first term is - 15, you have:

First term, A1 = - 15 + 9(0) = - 15
Second term, A2 =  - 15 + 9(1) = - 6
Third term, A3 = -15 + 9(2) = - 15 + 18 = 3
Fourth term, A4 = - 15 + 9(3) = - 15 + 27 = 12

3. So, the general formula is An = - 15 + 9 (n - 1), which is the option D)

3) Question 3. Find an equation for the nth term of the arithmetic sequence A14 = - 33, A15 = 9.

Answer: option B) An = - 579 + 42(n - 1)

Explanation:

1) Find the difference: 9 - (-33) = 9 + 33 = 42

2) A15 = A1 + 42 * (15 - 1)

=> A1 = A15 - 42(15 - 1)

A1 = A15 - 42(14)

A1 = 9 - 588 = - 579

Therefore, the formula es An = - 579 + 42(n - 1)

4) Question 4. Determine whether the sequence converges or diverges. If it converges, give the limit.

48, 8, 4/3, 2/9, ...

Answer: the sequence converges to 288/5

Explanation:

That is a geometric sequence.

The ratio is 1/6: 8/48 = 1/6; (4/3) / 8 = 4/24 = 1/6; (2/9)/(4/3) = 6/36 = 1/6.

The convergence criterium is that if |ratio| < 1 then the series, this is the sum of all the terms, converge to: A1 / (1 - ratio)

Then the limit 48 / (1 - 1/6) = 48 / (5/6) = 48*6 / 5 = 288/5

5) Question 5. Find an equation for the nth term of the sequence.

-3, -12, - 48, -192

Answer: - 3 * (4)^(n-1)

Explanation: clearly any term (from the second) is the previous term multiplied by 4.

The first term is  -3
The second term is -3(4) = - 12
The third term is -3(4)(4)= - 48
The fourth term is - 3 (4)(4)(4) = - 192

So, the general formula for the nth term is -3 * 4^ (n-1)

6) Question 6. Find an equation for the nth term of a geometric sequence where the second and fifth terms are -21 and 567, respectively.

Answer: An =7 * (-3)^(n-1)

Explanation:

1) The fith term is the second term * (ratio)^3: A5 = A3 * (r)^3

2)  A5 = 567, A2 = - 21 => r^3 = A5 / A2 = - 567 / 21 = - 27

=> r = ∛(-27) = - 3

3) So the first term is A1 = A2 / r = -21 / -3 = 7

4) The general formula is

An =7 * (-3)^(n-1)

7) Question 7. Write the sum using summation notation, assuming the suggested pattern continues.

5 - 15 + 45 - 135 + ...

Answer: option B) summation of five times negative three to the power of n from n equals zero to infinity

Explanation:

5 = 5
-15 = 5 (-3)
45 = 5(-3)^2
-135 = 5(-3)^3

=> 5 + 5(-3) + 5(-3)^2 + 5(-3)^3+....

Using the summation notation that is:

∞
∑ (5)(-3)^n
n=0

Which means summation of five times negative three to the power of n from n equals zero to infinity

8) Question 8. Write the sum using summation notation, assuming the suggested pattern continues.
-9 - 3 + 3 + 9 + ... + 81

Answer: option A) summation of the quantity negative nine plus six n from n equals zero to fifteen

Explanation:

Find the difference:

-3 - (-9) = - 3 + 9 = 6
3 - (-3) = 3 + 3 = 6
9 - 3 = 6

First term: - 9
Second term: - 9 + 6(1)
Third term: - 9 + 6(2)

nth term = - 9 + (n -1)

Summation = [- 9] + [- 9 + 6(1) + [-9 + 6(2)] + [-9 + 6(3) ]+ .... [-9 + 6(15) ]

Using summation notation:

15
∑ [-9 + 6n]
n=0

which means summation of the quantity negative nine plus six n from n equals zero to fifteen.


9) Question 9. Write the sum using summation notation, assuming the suggested pattern continues.


64 + 81 + 100 + 121 + ... + n2 + ...

Answer: A) summation of n squared from n equals eight to infinity

Explanation:

64 = 8^2
81 = 9^2
100 = 10^2121 = 11^2
n^2

=>
∞
∑ n^2
n=8

which means summation of n squared from n equals eight to infinity 10) Question 10. Find the sum of the arithmetic sequence.
17, 19, 21, 23, ..., 35

Answer: 260

Explanation:
The difference is 2:
The sum is: 17 + 19 + 21 + 23 + 25 + 27 + 29 + 31 + 33 + 35.
You can use the formula for the sum of an arithmetic sequence:

(A1 + An) * n / 2 = (17 + 35)*10/2 = 260


11) Question 11. Find the sum of the geometric sequence.  1, 1/2, 1/4, 1/8, 1/16Answer: option D) 31/16
Explanation:

You can either sum the 5 terms or use the formula for the partial sum of a geometric sequence.
The formula is: Sum = A * ( 1 - r^n) / (1 - r)

Here A = 1, r = 1/2, and n = 5 => Sum = 1 * (1 - (1/2)^5 ) / (1 - 1/2) =

= [ 1 - 1/32] / [1/2] = [31/32] / [1/2] = 31 / 16