The difference between the first term and the second term in a geometric sequence is 6. The difference between the second term and the third term is 3. Calculate the difference between the third term and the fourth term.
Thanks to anyone who helps
Let's just translate into Math:
"The difference between the first term and the second term in a geometric sequence is 6" -----> ar - a = 6
a(r-1) = 6 **
"The difference between the second term and the third term is 3. "
----> ar^2 - ar = 3
ar(r-1) = 3 ***
divide *** by **
ar(r-1) / (a(r-1)) = 3/6
r = 1/2
sub back into **
a(1/2-1) = 6
a = -12
the difference between the third term and the fourth term
= ar^3 - ar^2
= ar^2(r-1)
= -12(1/4)(-1/2) = 3/2
check: the sequence is -12, -6, -3, -3/2
find the differences as stated and you will see my answer is correct
To calculate the difference between the third term and the fourth term in a geometric sequence, we first need to determine the common ratio (r) for the sequence.
Let's denote the first term as a1, the second term as a2, the third term as a3, and the fourth term as a4.
We are given that the difference between the first term and the second term (a1 - a2) is 6, and the difference between the second term and the third term (a2 - a3) is 3.
Using this information, we can set up two equations to find the common ratio (r):
(a1 - a2) = 6 ----> Equation 1
(a2 - a3) = 3 ----> Equation 2
To solve these equations, we need to express the terms of the geometric sequence in terms of the first term (a1) and the common ratio (r):
a1 = a1 (first term)
a2 = a1 * r (second term)
a3 = a1 * r^2 (third term)
a4 = a1 * r^3 (fourth term)
Now we substitute these values into the equations:
(a1 - a1 * r) = 6 ----> Equation A (Replace a2 with a1 * r)
(a1 * r - a1 * r^2) = 3 ----> Equation B (Replace a3 with a1 * r^2)
Now, we can solve these two equations simultaneously to find the values of a1 and r:
From Equation A:
a1 - a1 * r = 6
a1(1 - r) = 6
a1 = 6 / (1 - r)
From Equation B:
a1 * r - a1 * r^2 = 3
a1 * r * (1 - r) = 3
Substitute the value of a1:
(6 / (1 - r)) * r * (1 - r) = 3
6 * r * (1 - r) = 3 * (1 - r)
Divide both sides by 3:
2 * r * (1 - r) = 1 - r
2r - 2r^2 = 1 - r
2r^2 - 3r + 1 = 0
We can now solve this quadratic equation for r using factoring or the quadratic formula. I'll use factoring in this explanation:
2r^2 - 3r + 1 = 0
(2r - 1)(r - 1) = 0
From this, we get two possible values for r:
2r - 1 = 0 --> r = 1/2
r - 1 = 0 --> r = 1
Now, we have two potential values for r, which means there are two possible geometric sequences that satisfy the given conditions.
To find the difference between the third term and the fourth term in each case, we substitute the values back into the equation for the fourth term (a4) and subtract the third term (a3) from it:
Case 1: r = 1/2
a4 = a1 * (1/2)^3 = a1/8
Difference = a4 - a3 = a1/8 - a1 * (1/2)^2 = a1/8 - a1/4 = -3a1/8
Case 2: r = 1
a4 = a1 * 1^3 = a1
Difference = a4 - a3 = a1 - a1 * 1^2 = a1 - a1 = 0
Therefore, in case 1, the difference between the third term and the fourth term is -3a1/8, where a1 represents the first term of the sequence. In case 2, the difference is 0.