In a GP, the sum of the 3rd and 4th terms is -4 /3, and the sum of the 4th and 5th terms is -4 /9. Find the 6th term
ar^2 + ar^3 = -4/3
ar^2(1 + r) = -4/3 **
ar^3 + ar^4 = -4/9
ar^3(1 + r) = -4/9 ***
divide *** by **
ar^3(1+r)/(ar^2(1+r)) = (-4/9) / (-4/3)
r = 1/3
plug back into **
a(1/3)^2 (1 + 1/3) = -4/3
a(1/9)(4/3) = -4/3
a = -9
term(6) = ar^5 = -9(1/3)^5 = -1/27
check: sequence is:
-9, -3, -1, -1/3, -1/9, -1/27, ...
is -1 - 1/3 = -4/3 ? , YES
is -1/3 - 1/9 = -4/9 ? YES
To find the 6th term of the geometric progression (GP), we need to determine the common ratio (r) first. Let's proceed:
Let the first term be a, and the common ratio be r.
The third term is given by: a * r^2
The fourth term is given by: a * r^3
The fifth term is given by: a * r^4
According to the problem, the sum of the third and fourth terms is -4/3. Hence, we can write the equation:
a * r^2 + a * r^3 = -4/3
Similarly, the sum of the fourth and fifth terms is -4/9, so we can write another equation:
a * r^3 + a * r^4 = -4/9
Let's solve these equations to find the values of a and r:
From the first equation, we can take a common factor:
a * r^2 * (1 + r) = -4/3
From the second equation, we can also take a common factor:
a * r^3 * (1 + r) = -4/9
Now, divide the second equation by three times the first equation:
(a * r^3 * (1 + r)) / (3 * a * r^2 * (1 + r)) = (-4/9) / (-4/3)
Simplifying the equation:
r / 3 = 1/3
r = 1
Now substitute the value of r into one of the original equations:
a * (1^2) + a * (1^3) = -4/3
2a = -4/3
a = -2/3
Now that we have the values of a and r, we can find the 6th term:
The 6th term is given by: a * r^5
Substituting the values we found:
(-2/3) * (1^5) = -2/3
Therefore, the 6th term of the GP is -2/3.