1. Find all values of p so that 6p, 4p-1, p^2 - 1 will be an arithmetic sequence.
2. The common ration of a geometric sequence is -3. IF the terms of this sequence are 3y, y+6, z+1, ..... respectively, what is the value of z?
#1: (4p-1)-(6p) = (p^2-1)-(4p-1)
#2:
(y+6)/3y = -3
y = -3/5
(z+1)/(y+6) = -3
Now plug in y and solve for z
1. To determine the values of p for which 6p, 4p-1, and p^2-1 form an arithmetic sequence, we need to find the common difference between consecutive terms.
In an arithmetic sequence, the common difference is the constant value that is added or subtracted to obtain the next term.
The common difference (d) can be found by subtracting the second term from the first term. In this case, we have:
(4p-1) - (6p) = -2p - 1
Next, we subtract the second term (4p-1) from the third term (p^2-1) to check if it is equal to the common difference (d). We have:
(p^2-1) - (4p-1) = p^2 - 4p
Since we want the terms to form an arithmetic sequence, we set -2p - 1 equal to p^2 - 4p and solve for p.
-2p - 1 = p^2 - 4p
Rearrange the equation and set it equal to zero:
p^2 + 2p - 4p + 1 = 0
p^2 - 2p + 1 = 0
Factor the quadratic equation:
(p - 1)^2 = 0
Take the square root on both sides:
p - 1 = 0
p = 1
Therefore, the value of p that makes the terms 6p, 4p-1, and p^2-1 an arithmetic sequence is p = 1.
2. The problem states that the common ratio of the geometric sequence is -3 and the terms are given by 3y, y+6, z+1, and so on.
In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio. In this case, the common ratio is -3.
To determine the value of z, we need to find the third term (z+1) in terms of the first term (3y).
The first term (3y) multiplied by the common ratio (-3) equals the second term (y+6). We can set up the equation as follows:
3y * -3 = y + 6
-9y = y + 6
To solve for y, we can simplify the equation:
-9y - y = 6
-10y = 6
y = -6/10
y = -3/5
Now, substitute the value of y back into the equation for the second term to find z:
y + 6 = -3/5 + 6 = -3/5 + 30/5 = 27/5
Therefore, the value of z is 27/5.