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.
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
√33π= 5.7445626
Why did the geometric progression go to see a therapist? Because it had a lot of negative terms and needed to work through its issues.
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!
To find the 6th term of a geometric progression (GP), we need to determine the common ratio (r) and the first term (a1).
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.