1) Determine the value of r (put this number in the first blank) and then determine the value of the 10th term of the geometric sequence. Do not use any decimals or mixed numbers in your answers. Reduce all fractions to lowest terms.
a1 = 2 and a4 = −1/4
2)Find d and then find the 20th term of each sequence. Type the value of d (just the number) in the first blank and then type the 20th term(just the number) in the second blank.
a1 = 10 and a3 = 28
3)Find d and then find the 20th term of each sequence. Type the value of d (just the number) in the first blank and then type the 20th term(just the number) in the second blank.
a5 = -13 and a12 = -27
To determine the value of r and the 10th term of the geometric sequence in question 1, we can use the formula for the nth term of a geometric sequence:
an = a1 * r^(n-1)
First, let's find the value of r:
Since a1 = 2 and a4 = -1/4, we can use the formula to create two equations:
a4 = a1 * r^(4-1)
-1/4 = 2 * r^3
Now, let's solve this equation for r:
-1/4 = 2 * r^3
Divide both sides by 2:
-1/8 = r^3
Take the cube root of both sides:
r = -1/2
Now that we have the value of r, we can find the 10th term (a10) using the formula:
a10 = a1 * r^(10-1)
a10 = 2 * (-1/2)^9
Simplifying:
a10 = 2 * (-1/2)^9
a10 = 2 * (-1/2)^8 * (-1/2)
a10 = 2 * 1/256 * (-1/2)
a10 = -1/256
Therefore, the value of r is -1/2 and the 10th term of the geometric sequence is -1/256.
Now, let's move on to question 2:
We are given a1 = 10 and a3 = 28. To find the common difference (d), we can use the formula for the nth term in an arithmetic sequence:
an = a1 + (n-1) * d
Using the given values:
a1 + 2d = a3
Substituting the values:
10 + 2d = 28
Solving for d:
2d = 28 - 10
2d = 18
d = 9
Now that we have the value of d, we can find the 20th term (a20) using the formula:
a20 = a1 + (20-1) * d
a20 = 10 + 19 * 9
a20 = 10 + 171
a20 = 181
Therefore, the value of d is 9 and the 20th term of the arithmetic sequence is 181.
Finally, let's solve question 3:
We are given a5 = -13 and a12 = -27. To find the common difference (d), we can use the formula for the nth term in an arithmetic sequence:
an = a1 + (n-1) * d
Using the given values:
a1 + 4d = a5
a1 + 11d = a12
Substituting the values:
a1 + 4d = -13
a1 + 11d = -27
Now we can solve this system of equations to find the values of a1 and d:
Multiply the first equation by 11:
11a1 + 44d = -143
Multiply the second equation by 4:
4a1 + 44d = -108
Subtract the second equation from the first:
11a1 + 44d - (4a1 + 44d) = -143 - (- 108)
7a1 = -35
a1 = -5
Now that we have the value of a1, we can substitute it into the equation a1 + 4d = -13 to find the value of d:
(-5) + 4d = -13
4d = -13 + 5
4d = -8
d = -2
Now that we have the value of d, we can find the 20th term (a20) using the formula:
a20 = a1 + (20-1) * d
a20 = -5 + 19 * (-2)
a20 = -5 - 38
a20 = -43
Therefore, the value of d is -2 and the 20th term of the arithmetic sequence is -43.
To solve these problems, we need to use the formulas for finding the nth term of a geometric sequence and the common ratio.
1) In a geometric sequence, each term is found by multiplying the previous term by a constant number called the common ratio (r). We are given that a1 = 2 and a4 = -1/4. The general formula for the nth term (an) of a geometric sequence is: an = a1 * r^(n-1).
If we substitute n = 4, we can solve for the common ratio (r) using the given information:
a4 = a1 * r^(4-1)
-1/4 = 2 * r^3
Simplifying the equation:
r^3 = (-1/4) / 2
r^3 = -1/8
Cubing both sides:
r = ∛(-1/8)
r = -1/2
So, the value of r is -1/2.
To find the 10th term, we can now substitute n = 10 into the formula:
a10 = a1 * r^(10-1)
a10 = 2 * (-1/2)^(9)
a10 = 2 * (-1/512)
a10 = -1/256
Therefore, the 10th term of the geometric sequence is -1/256.
2) Similar to the previous problem, we are given a1 = 10 and a3 = 28. We need to find the common difference (d) and then determine the 20th term.
For an arithmetic sequence, the nth term (an) is given by the formula: an = a1 + (n-1)*d.
Given a1 = 10 and a3 = 28, we can use these values to find the common difference (d).
Substituting n = 3 into the formula, we get:
a3 = a1 + (3-1)*d
28 = 10 + 2d
18 = 2d
d = 18/2
d = 9
So, the value of d is 9.
To find the 20th term, we can substitute n = 20 into the formula:
a20 = a1 + (20-1)*d
a20 = 10 + 19*9
a20 = 10 + 171
a20 = 181
Therefore, the 20th term of the arithmetic sequence is 181.
3) Similarly, we are given a5 = -13 and a12 = -27. We need to find the common difference (d) and then determine the 20th term.
Using the arithmetic sequence formula, we can find the common difference (d) by substituting n = 5 and n = 12:
a5 = a1 + (5-1)*d
-13 = a1 + 4d
a12 = a1 + (12-1)*d
-27 = a1 + 11d
We have a system of equations here:
a1 + 4d = -13 ----(1)
a1 + 11d = -27 ----(2)
Subtracting equation (1) from equation (2), we can eliminate a1:
a1 + 11d - (a1 + 4d) = -27 - (-13)
7d = -14
d = -14/7
d = -2
So, the value of d is -2.
Now, to find the 20th term, we can substitute n = 20 into the formula:
a20 = a1 + (20-1)*d
a20 = a1 + 19*(-2)
a20 = a1 - 38
Since we don't know the exact value of a1, we cannot determine the 20th term without additional information.