Find the 7th term of geometric progression if the second term is 75, the 5th term,3/5.
using our definitions,
ar = 75
ar^4 = 3/5
divide the 2nd by the 1st:
r^3 = 3/375 = 1/125
r = 1/5
back in ar = 75
a(1/5) = 75
a = 375
term7 = ar^6
= 375(1/15625)
= 3/125
check my arithmetic
To find the 7th term of a geometric progression, we need to determine the common ratio (r) first.
We are given that the second term is 75 and the fifth term is 3/5. Let's denote the first term as a.
The formula for the nth term of a geometric progression is:
An = a * r^(n-1)
Using this formula, we can set up two equations to solve for the common ratio (r).
Equation 1: a * r = 75 (since the second term, Ar, is equal to 75)
Equation 2: a * r^4 = 3/5 (since the fifth term, A5, is equal to 3/5)
To eliminate the variable 'a', we can divide Equation 2 by Equation 1:
(a * r^4) / (a * r) = (3/5) / 75
Simplifying this equation, we get:
r^3 = (3/5) / 75
r^3 = 3 / (5 * 75)
r^3 = 1 / 125
r = (1 / 125)^(1/3)
r = 1 / 5
Now that we have found the value of the common ratio (r = 1/5), we can find the first term (a) by substituting into Equation 1:
a * (1/5) = 75
a = 75 * 5
a = 375
With the values of a = 375 and r = 1/5, we can find the 7th term (A7) using the formula for the nth term:
A7 = a * r^(7-1)
A7 = 375 * (1/5)^6
A7 = 375 * (1/5)^6
A7 = 375 * (1/15625)
A7 = 375/15625
A7 = 3/125
Therefore, the 7th term of the geometric progression is 3/125.