To find the 12th term of a geometric sequence, we first need to identify the relationship between the terms. A geometric sequence can be described by the formula for its nth term, which is:
an = a1 * r(n-1)
Where:
- an is the nth term,
- a1 is the first term,
- r is the common ratio, and
- n is the term number.
We are given:
- a1 = 8
- a6 = 8192
First, we can write the sixth term using the formula:
a6 = a1 * r(6-1)
Substituting the known values:
8192 = 8 * r5
To isolate r5, divide both sides by 8:
r5 = 8192 / 8
Calculating the right side gives:
r5 = 1024
Then, we need to find the value of r by taking the fifth root of both sides:
r = 1024(1/5)
Since 1024 is equal to 210, we can write:
r = (210)(1/5) = 2(10/5) = 22 = 4
Now that we have the common ratio (r = 4), we can find the 12th term:
a12 = a1 * r(12-1)
Substituting the known values:
a12 = 8 * 411
Now we need to calculate 411. Knowing that 4 = 22, we can write:
411 = (22)11 = 2(2*11) = 222
So, substituting this back in:
a12 = 8 * 222
Since 8 is equal to 23, we have:
a12 = 23 * 222 = 2(3 + 22) = 225
Thus, the 12th term of the geometric sequence is:
a12 = 33554432
In conclusion, the 12th term of the geometric sequence where the first term is 8 and the sixth term is 8192 is 33554432.