I have 3 questions I need help with.

1) what is the 15th term of the sequence 81,27,9,...

2) what is a possible value for the missing term of the geometric sequence 39,_,975,_

3) what is the sum of the finite arithmetic series (-5)+0+5+10+..+65

1. proof that my answer is correct

81 27 9 3 1 1/3 1/9 1/27 1/81 1/243 1/729 1/2187 1/6561 1/19683 1/59049
count the terms

2. proof my answer is correct:
195/39 = 5
975/195 = 5

-195/39 - -5
975/-195 = -5

your first sequence looks geometric with

a = 81, r = 1/3

term(15) = ar^14
= 81(1/4782969)
= 1/59049

2. let the missing term be x
then x/39 = 975/x
x^2 = 38025
x = ± √38025 = ± 195
could be 195 or -195

3. looks arithmetic with a = -5 , d = 5
term(n) = 65
-5 + (n-1)(5) = 65
5n - 5 = 70
5n = 75
so we have 15 terms

sum(15) = (15/2)(first + last)
= 15/2(-5+65) = 450

the first and second is not a possible answer.

1) 1/729
1/243
1/81
1/810

Sure, I can help you with those questions! Let's take them one by one.

1) To find the 15th term of the sequence 81, 27, 9,..., we need to identify the pattern of the sequence first. Looking closely, we can see that each term is obtained by dividing the previous term by 3. So, the sequence is decreasing geometrically with a common ratio of 1/3.

To find the 15th term, we can use the formula for a geometric sequence: an = a1 * r^(n-1), where an is the nth term, a1 is the first term, r is the common ratio, and n is the term number.

Given that a1 = 81 and r = 1/3, we can substitute these values into the formula to find the 15th term:
a15 = 81 * (1/3)^(15-1)
a15 = 81 * (1/3)^14
a15 ≈ 0.027

Therefore, the 15th term of the sequence is approximately 0.027.

2) To find a possible value for the missing term in the geometric sequence 39, _, 975, _, we need to identify the common ratio between the terms. To do this, we can take any two consecutive terms and divide the larger term by the smaller term.

Let's consider the ratio of the second term to the first term:
27/81 = 1/3

Now, let's find the ratio of the third term to the second term:
9/27 = 1/3

Since both ratios are equal, we can conclude that the common ratio in this sequence is 1/3.

To find the missing term, we can use the formula for a geometric sequence: an = a1 * r^(n-1), where an is the nth term, a1 is the first term, r is the common ratio, and n is the term number.

Given that a1 = 39, r = 1/3, and n could be either 2 or 4 (since there are two missing terms), we can substitute these values into the formula.

For n = 2:
a2 = 39 * (1/3)^(2-1)
a2 = 39 * (1/3)
a2 = 13

For n = 4:
a4 = 39 * (1/3)^(4-1)
a4 = 39 * (1/3)^3
a4 = 39 * (1/27)
a4 ≈ 1.444

Therefore, a possible value for the missing term in the sequence is either 13 or approximately 1.444.

3) To find the sum of the finite arithmetic series (-5) + 0 + 5 + 10 + ... + 65, we can use the formula for the sum of an arithmetic series:

Sn = (n/2) * (a1 + an),

where Sn is the sum of the series, n is the number of terms, a1 is the first term, and an is the nth term.

First, let's find the number of terms (n). The last term, an, is 65, and each term increases by 5. So, we can calculate the number of terms by finding the common difference (d) and using the formula:

an = a1 + (n-1)d.

Using the formula for arithmetic sequences, we can rewrite an as:

65 = (-5) + (n-1) * 5
65 = -5 + 5n - 5
65 = 5n - 10
75 = 5n
n = 15.

Therefore, there are 15 terms in the arithmetic series.

Now, let's calculate the sum (Sn):

Sn = (n/2) * (a1 + an)
Sn = (15/2) * (-5 + 65)
Sn = (15/2) * 60
Sn = 450.

Therefore, the sum of the arithmetic series is 450.

I hope this helps! Let me know if you have any further questions.