a, ar ar^2 ar^3 ar^4
a r^2 + a r^3 = -4/3
a r^3 + a r^4 = -4/9
a r^2 (1 + r) = -4/3
a r^3 (1 + r) = -4/9
-4/(3 a r^2) = -4/(9 a r^3)
so
r = 1/3
a(1/9)(4/3) = -4/3
a = -9
so:
-9 -3 -1 -1/3 -1/9 and -1/27
a r^2 + a r^3 = -4/3
a r^3 + a r^4 = -4/9
a r^2 (1 + r) = -4/3
a r^3 (1 + r) = -4/9
-4/(3 a r^2) = -4/(9 a r^3)
so
r = 1/3
a(1/9)(4/3) = -4/3
a = -9
so:
-9 -3 -1 -1/3 -1/9 and -1/27
To find the 6th term, let's call the first term "a" and the common ratio "r".
The 3rd term is ar^2, the 4th term is ar^3, and the 5th term is ar^4.
We know that the sum of the 3rd and 4th terms is -4/3, so ar^2 + ar^3 = -4/3.
We also know that the sum of the 4th and 5th terms is -4/9, so ar^3 + ar^4 = -4/9.
To find the 6th term, we need to solve the equation ar^4 + ar^5.
Let's solve for "r" first by dividing the equation ar^2 + ar^3 = -4/3 by "ar^2".
This gives us 1 + r = (-4/3) / (ar^2).
Simplifying further, we get 1 + r = -4 / (3ar^2).
Now, let's solve for "r" in the second equation, ar^3 + ar^4 = -4/9. Dividing by "ar^3", we get 1 + r = (-4/9) / (ar^3).
Simplifying this, we have 1 + r = -4 / (9ar^3).
Since both equations are equal to 1 + r, we can set them equal to each other:
-4 / (3ar^2) = -4 / (9ar^3).
Now, we can cancel out the common factors and simplify the equation to:
3r = r^2.
Dividing both sides by "r" (assuming r is not 0), we are left with:
3 = r.
Now that we know the common ratio "r" is 3, we can substitute it back into one of the original equations to find the value of "a".
Let's use ar^2 + ar^3 = -4/3.
Substituting 3 for "r", we get a(3^2) + a(3^3) = -4/3.
Simplifying this further, we have 9a + 27a = -4/3.
Combining like terms, we get 36a = -4/3.
Dividing both sides by 36, we find that a = -1/27.
Now, we have the values of "a" and "r".
The 6th term is given by ar^5, so plugging in the values, we have:
(-1/27)(3^5) = -1/27 * 243 = -9.
So, the 6th term of the geometric progression is -9.
Hope you found this arithmetic a-musing!
Let's assume the first term of the GP is 'a1', and the common ratio is 'r'.
The general formula to find the nth term of a GP is given by:
an = a1 * r^(n-1)
Now, let's use the given information to form two equations:
Equation 1: The sum of the 3rd and 4th terms is -4/3.
a3 + a4 = -4/3
We substitute the values using the general formula:
a1 * r^(3-1) + a1 * r^(4-1) = -4/3
a1 * r^2 + a1 * r^3 = -4/3
Equation 2: The sum of the 4th and 5th terms is -4/9.
a4 + a5 = -4/9
Again, substituting the values using the general formula:
a1 * r^(4-1) + a1 * r^(5-1) = -4/9
a1 * r^3 + a1 * r^4 = -4/9
Now, we have two equations:
a1 * r^2 + a1 * r^3 = -4/3 -- (Equation 1)
a1 * r^3 + a1 * r^4 = -4/9 -- (Equation 2)
We can solve these equations simultaneously to find the values of a1 and r.
To do this, we can divide Equation 1 by Equation 2 to eliminate a1:
(a1 * r^2 + a1 * r^3) / (a1 * r^3 + a1 * r^4) = (-4/3) / (-4/9)
Simplifying further:
(r^2 + r^3) / (r^3 + r^4) = (-4/3) / (-4/9)
(r^2 + r^3) * (9 / (r^3 + r^4)) = -3
Cross-multiplying:
(r^2 + r^3) * 9 = -3 * (r^3 + r^4)
9r^2 + 9r^3 = -3r^3 - 3r^4
Rearranging the terms:
3r^4 + 12r^3 + 9r^2 = 0
Simplifying further by dividing all terms by 3:
r^4 + 4r^3 + 3r^2 = 0
Now, we can factor this expression:
r^2 (r + 1) (r + 3) = 0
From this factorization, we have three possible values for 'r': r = 0, r = -1, and r = -3.
Next, we substitute each of these values back into any of our earlier equations (Equation 1 or Equation 2) to solve for 'a1'.
For example, let's substitute r = 0 into Equation 1:
a1 * (0)^2 + a1 * (0)^3 = -4/3
0a1 + 0a1 = -4/3
0 = -4/3
This equation yields an invalid result, which means r = 0 is not the correct value.
Now, substitute r = -1 into Equation 1:
a1 * (-1)^2 + a1 * (-1)^3 = -4/3
a1 - a1 = -4/3
0 = -4/3
Again, this equation yields an invalid result, so r = -1 is not the correct value either.
Lastly, substitute r = -3 into Equation 1:
a1 * (-3)^2 + a1 * (-3)^3 = -4/3
9a1 + 27a1 = -4/3
36a1 = -4/3
a1 = (-4/3) / 36
a1 = -1/27
So, for r = -3, we have found the value of 'a1' as -1/27.
Now, let's use the value of 'a1' and 'r' we found to determine the 6th term (a6) using the general formula:
a6 = a1 * r^(6-1)
a6 = (-1/27) * (-3)^(6-1)
a6 = (-1/27) * (-3)^5
a6 = (-1/27) * (-243)
a6 = 9
Hence, the 6th term of the geometric progression is 9.