The first, third and ninth terms of an arithmetic sequence form the terms of a geometric sequence. Find the common ratio of the geometric sequences.
a, a+2d, a+8d
a+2d/a = a+8d/a+2d
?
great so far:
(a+2d)/a = (a+8d)/(a+2d)
(a+2d)^2 = a(a+8d)
a^2 + 4ad + 4d^2 = a^2 + 8ad
4d^2 = 4ad
4d = 4a
a = d
r = (a+2d)/a
= (d + 2d)/d
= 3d/d
= 3
makes sense to me:
looks like we can start with any value for a
e.g let a = 5
then d = 5
and for the AS
t1 = 5
t3 = 5+10 = 15
t9 = 5 + 8(5) = 45
and 5, 15, 45 are in a GS, with r = 3
Correct
To find the common ratio of the geometric sequence formed by the first, third, and ninth terms of an arithmetic sequence, we need to set up an equation using the terms of the arithmetic sequence.
The first term of the arithmetic sequence is "a."
The third term of the arithmetic sequence is "a + 2d."
The ninth term of the arithmetic sequence is "a + 8d."
Let's set up the equation using these terms:
(a + 2d) / a = (a + 8d) / (a + 2d)
Now, let's solve this equation to find the value of the common ratio.
First, we can cross-multiply:
(a + 2d)(a + 2d) = (a + 8d)(a)
Expanding both sides gives:
a^2 + 4ad + 4d^2 = a^2 + 8ad
Next, simplify the equation by canceling out the a^2 terms:
4ad + 4d^2 = 8ad
Now, subtract 4ad from both sides:
4d^2 = 8ad - 4ad
Simplifying further:
4d^2 = 4ad
Divide both sides by 4d:
d = a
Therefore, since d = a, the common ratio of the geometric sequence is 1.
To find the common ratio of the geometric sequence formed by the first, third, and ninth terms of the arithmetic sequence, we can use the formula for the nth term of an arithmetic sequence and the formula for the nth term of a geometric sequence.
The nth term of an arithmetic sequence can be expressed as: a + (n - 1)d,
where a is the first term and d is the common difference.
In this case, the first term of the arithmetic sequence is a, and the third term is a + 2d. Plugging these values into the formula for the nth term of an arithmetic sequence, we get:
a + 2d = a + (3 - 1)d
a + 2d = a + 2d
This equation tells us that the third term is equal to itself, which is always true for an arithmetic sequence.
Now, let's consider the geometric sequence formed by the first, third, and ninth terms. In a geometric sequence, each term is obtained by multiplying the previous term by a common ratio.
The first term of the geometric sequence is a, the second term is a + 2d (which corresponds to the third term of the arithmetic sequence), and the third term is a + 8d.
Therefore, we have the equation:
(a + 2d) / a = (a + 8d) / (a + 2d)
To find the common ratio, we can cross-multiply and solve the resulting equation:
(a + 2d)(a + 2d) = a(a + 8d)
Expanding and simplifying this equation gives:
a^2 + 4ad + 4d^2 = a^2 + 8ad
Cancelling out the terms that appear on both sides and moving all terms to one side, we get:
4ad + 4d^2 - 8ad = 0
Simplifying further:
-4ad + 4d^2 = 0
Factoring out a common factor of 4d, we have:
4d(-a + d) = 0
Since the product of two factors is zero, either 4d = 0 or -a + d = 0.
From 4d = 0, we deduce that d = 0. However, since we are looking for a non-zero common ratio for the geometric sequence, we can discard this solution.
Therefore, we can set -a + d = 0 and solve for d:
-a + d = 0
d = a
The common ratio of the geometric sequence is equal to the common difference of the arithmetic sequence.