The 1st term of a geometric sequence is 25 and the 6th term is -1/125. What is the 3rd term?
using the formulas for GP's
a = 25
ar^5 = -1/125
r^5 = (-1/125)(1/25) = -1/3125
r = (-1/3125)^(1/5) = -1/5
term3 = ar^2 = 25(-1/5)^2 = 1
To find the 3rd term of a geometric sequence, you need to know the common ratio. The common ratio (r) is found by dividing any term by its previous term.
Given that the 1st term is 25 and the 6th term is -1/125, we can find the common ratio by dividing the 6th term by the 5th term.
The 5th term can be found by dividing the 6th term by the common ratio: -1/125 = (25 * r^4)/(25 * r^5), where r is the common ratio.
Now, simplify the equation: -1/125 = 1/(r * 5)
Multiply both sides of the equation by r * 5 to isolate r: -r/125 = 1
Multiply both sides of the equation by -125 to solve for r: r = -125
Now that we have the common ratio (r), we can find the 3rd term by multiplying the 1st term (25) by r^2 (r raised to the power of 2).
Therefore, the 3rd term is 25 * (-125)^2 = 25 * 15625 = 390,625.