in a certain geometric sequence, the third term is 8 and the sixth term is 125. what is the first term?
Reiny already answered this for you.
http://www.jiskha.com/display.cgi?id=1311380020
To find the first term of a geometric sequence, we can use the formula for the nth term of a geometric sequence, which is given by:
a_n = a_1 * r^(n-1),
where a_n is the nth term, a_1 is the first term, r is the common ratio, and n is the position of the term in the sequence.
In this case, we are given that the third term (n = 3) is 8, and the sixth term (n = 6) is 125. Plugging these values into the formula, we can set up two equations:
8 = a_1 * r^(3-1),
125 = a_1 * r^(6-1).
Dividing the second equation by the first equation, we can eliminate a_1, giving us:
125/8 = r^(6-1)/(r^(3-1)),
125/8 = r^5 / r^2.
Simplifying the equation further, we get:
125/8 = r^3.
To solve for r, we take the cube root of both sides:
(125/8)^(1/3) = r.
Now, we can substitute the value of r into one of the previous equations to find a_1. Let's use the first equation:
8 = a_1 * (125/8)^(1/3)^(3-1).
Simplifying the right side, we get:
8 = a_1 * (125/8)^(1/3)^2,
8 = a_1 * (125/8)^(2/3).
To evaluate (125/8)^(2/3), we can take the cube root of both the numerator and the denominator:
8 = a_1 * (5/2)^2,
8 = a_1 * (25/4).
To isolate a_1, we can multiply both sides by 4/25:
8 * 4/25 = a_1,
a_1 = 32/25.
Therefore, the first term of the geometric sequence is 32/25.