the 4th term of a geometric sequence is 1/2 and the tenth term is 1/128 find the 10th term?
You have already said what the tenth term is.
To find the 10th term of a geometric sequence, we need to use the formula for the nth term of a geometric sequence:
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.
Given information:
- 4th term (a4) = 1/2
- 10th term (a10) = 1/128
Let's solve for the common ratio (r) first:
a4 = a1 * r^(4-1)
1/2 = a1 * r^3
Similarly, for the 10th term:
a10 = a1 * r^(10-1)
1/128 = a1 * r^9
Now, we can set up a system of equations to find a1 and r:
1/2 = a1 * r^3 -- Equation (1)
1/128 = a1 * r^9 -- Equation (2)
Divide Equation (2) by Equation (1) to eliminate a1:
(1/128) / (1/2) = (a1 * r^9) / (a1 * r^3)
1/128 * 2/1 = r^(9-3)
1/64 = r^6
Solving for r:
r^6 = 1/64
Taking the sixth root of both sides:
r = (1/64)^(1/6)
Simplifying:
r = 1/2
Now that we know the common ratio (r = 1/2), we can substitute it back into Equation (1) to solve for a1:
1/2 = a1 * (1/2)^3
1/2 = a1 * 1/8
Multiply both sides by 8/1 to isolate a1:
a1 = (1/2) * (8/1)
a1 = 4
Now we have the values of a1 = 4 and r = 1/2. We can substitute these into the formula to find the 10th term (a10):
a10 = a1 * r^(10-1)
a10 = 4 * (1/2)^(10-1)
a10 = 4 * (1/2)^9
a10 = 4 * (1/2)^8 * (1/2)
a10 = 4 * (1/256) * (1/2)
a10 = 4/512
a10 = 1/128
Therefore, the 10th term of the geometric sequence is 1/128.