The 6th term of a geometric sequence is 16 and the 3rd term is 2. find the first term and the common ratio.
To find the first term and the common ratio of a geometric sequence, we can use the formula for the nth term of a geometric sequence:
an = a1 * r^(n-1)
where:
an = nth term
a1 = first term
r = common ratio
n = position of the term
Given that the 3rd term is 2, we can substitute these values into the formula to get:
2 = a1 * r^(3-1)
2 = a1 * r^2
Similarly, for the 6th term:
16 = a1 * r^(6-1)
16 = a1 * r^5
Now we have a system of equations:
2 = a1 * r^2
16 = a1 * r^5
One way to solve this system of equations is by substitution. We can isolate a1 in the first equation:
a1 = 2 / r^2
Substitute this expression for a1 into the second equation:
16 = (2 / r^2) * r^5
Simplify:
16 = 2r^3
Divide both sides by 2:
8 = r^3
Take the cube root of both sides:
r = 2
Now that we have the value of r, we can substitute it into one of the original equations to find a1:
2 = a1 * r^2
2 = a1 * 2^2
2 = 4a1
Divide both sides by 4:
a1 = 1/2
Therefore, the first term (a1) is 1/2 and the common ratio (r) is 2.