the 12th term in a geometric sequence is 49, and the 14th term is 2401 . Find the 15th term
just use the formulas you have:
T12 = ar^11 = 49
T14 = ar^13 = 2401
T14/T12 = r^2 = 49
so, r = 7 or -7
T15 = T14 * r = 16807 or -16807.
Can't tell, since we don't know whether r is 7 or -7.
To find the 15th term of a geometric sequence, we need to find the common ratio first.
The formula for the nth term of a geometric sequence is:
an = a1 * r^(n-1)
Given that the 12th term (a12) is 49, and the 14th term (a14) is 2401:
a12 = a1 * r^(12-1) -> 49 = a1 * r^11
a14 = a1 * r^(14-1) -> 2401 = a1 * r^13
To eliminate a1, we can divide the second equation by the first equation:
(2401 / 49) = (a1 * r^13) / (a1 * r^11)
49 = r^2
Taking the square root of both sides:
r = ±√49
r = ±7
Since we are looking for the common ratio, we can ignore the negative value of -7, and take r = 7.
Now that we have the common ratio, we can find the 15th term (a15) using the formula:
a15 = a1 * r^(15-1)
We know that a12 = 49, so:
49 = a1 * 7^11
To find a1, we can divide both sides by 7^11:
(49 / 7^11) = a1
Now, substitute this value for a1 in the formula for a15:
a15 = (49 / 7^11) * 7^14
Simplifying:
a15 = 49 * 7^3
Calculating this:
a15 = 49 * 343
a15 = 16,807
Therefore, the 15th term of the geometric sequence is 16,807.
To find the 15th term of the geometric sequence, we first need to determine the common ratio of the sequence.
The formula for calculating the nth term \(a_n\) of a geometric sequence is as follows:
\[a_n = a_1 * r^{(n-1)},\]
where \(a_1\) is the first term and \(r\) is the common ratio.
We are given that the 12th term (\(a_{12}\)) is 49 and the 14th term (\(a_{14}\)) is 2401. We can use these values to set up two equations:
\[49 = a_1 * r^{(12-1)} \quad \text{(1)}\]
\[2401 = a_1 * r^{(14-1)} \quad \text{(2)}\]
Now, we can divide equation (2) by equation (1) to eliminate the \(a_1\) term:
\[\frac{{2401}}{{49}} = \frac{{a_1 * r^{13}}}{{a_1 * r^{11}}}\]
Simplifying, we get:
\[49 = r^2\]
Taking the square root of both sides gives us:
\[r = \sqrt{49} = 7\]
Now that we know the common ratio \(r = 7\) of the geometric sequence, we can find the first term (\(a_1\)). Plugging the value of \(r\) into equation (1):
\[49 = a_1 * 7^{(12-1)}\]
Simplifying:
\[49 = a_1 * 7^{11}\]
Dividing both sides by \(7^{11}\):
\[\frac{{49}}{{7^{11}}} = a_1\]
Evaluating this expression:
\[a_1 = \frac{{49}}{{7^{11}}} = \frac{{49}}{{19487171}}\]
Now that we have the value of the first term \(a_1\) and the common ratio \(r\), we can calculate the 15th term (\(a_{15}\)) using the formula for the nth term:
\[a_{15} = a_1 * r^{(15-1)}\]
Plugging in the values we found:
\[a_{15} = \frac{{49}}{{19487171}} * 7^{14}\]
Evaluating this expression will give us the 15th term of the geometric sequence.