in a certain geometric sequence, the third term is 8 and the sixth term is 125. what is the first term?
ar^2 = 8
ar^5 = 125
divide the 2nd by the 1st ....
r^3 = 128/8
r = 5/2
back into the first
a(25/4) = 8
etc.
To find the first term of the geometric sequence, we need to understand the relationship between the terms and the common ratio.
Let's denote the first term as 'a' and the common ratio as 'r'. The n-th term of a geometric sequence can be calculated using the formula:
tn = a * r^(n-1)
Given that the third term is 8, we can write this as:
8 = a * r^(3-1)
8 = a * r^2
Similarly, we can write the equation for the sixth term as:
125 = a * r^(6-1)
125 = a * r^5
Now we have a system of two equations:
Equation 1: 8 = a * r^2
Equation 2: 125 = a * r^5
To find the value of 'a', we can solve these equations simultaneously.
Dividing Equation 2 by Equation 1, we get:
125/8 = (a * r^5) / (a * r^2)
15.625 = r^3
Now we need to find the cube root of 15.625, which is approximately 2.5.
Substituting this value back into Equation 1, we can solve for 'a':
8 = a * (2.5)^2
8 = a * 6.25
Dividing both sides by 6.25, we get:
a = 8 / 6.25
a = 1.28
Therefore, the first term of the geometric sequence is approximately 1.28.