first three terms of a geometric sequence are the same as the 1st, 9th and 11th terms of an arithmetic sequence, respectively. What is the common ratio in the geometric sequence ? Thanks!
To find the common ratio in the geometric sequence, we need to compare the first three terms of the geometric sequence with the corresponding terms of the arithmetic sequence.
Let's assume the first term of the arithmetic sequence is 'a' and the common difference is 'd'.
The 1st term in the arithmetic sequence is 'a'.
The 9th term in the arithmetic sequence is 'a + (8d)'.
The 11th term in the arithmetic sequence is 'a + (10d)'.
Now, since the first three terms of the geometric sequence are the same as the 1st, 9th, and 11th terms of the arithmetic sequence respectively, we can set up the following equations:
Term 1 of the geometric sequence = Term 1 of the arithmetic sequence
Term 2 of the geometric sequence = Term 9 of the arithmetic sequence
Term 3 of the geometric sequence = Term 11 of the arithmetic sequence
Let's start by comparing the first terms:
Term 1 of the geometric sequence = Term 1 of the arithmetic sequence
This means that the first term of the geometric sequence is 'a'.
Next, let's compare the second terms:
Term 2 of the geometric sequence = Term 9 of the arithmetic sequence
This means that the second term of the geometric sequence is 'a + (8d)'.
Finally, let's compare the third terms:
Term 3 of the geometric sequence = Term 11 of the arithmetic sequence
This means that the third term of the geometric sequence is 'a + (10d)'.
To find the common ratio in the geometric sequence, we can divide the second term by the first term and the third term by the second term.
Common ratio = (Second term / First term) = ((a + (8d)) / a)
Common ratio = (Third term / Second term) = ((a + (10d)) / (a + (8d)))
Since the common ratio remains the same for all terms in a geometric sequence, we can equate these two ratios:
(a + (8d)) / a = (a + (10d)) / (a + (8d))
Cross multiply:
(a + (8d))(a + (8d)) = a(a + (10d))
Expand and simplify:
a^2 + 8ad + 8ad + 64d^2 = a^2 + 10ad
Simplify further:
64d^2 = 2ad
Divide both sides by '2d':
32d = a
So, the common ratio in the geometric sequence is 32.