the third and the sixth term of a G.p are 5 and -40 respectively. find the tenth term of the sequence?
using the definitions of a GP
ar^2 = 5
ar^5 = -40
divide the second by the first
r^3 = -8
r = -2
back in the first:
ar^2 = 5 ---> a(4) = 5
a = 5/4
term(10) = ar^9
= (5/4)(-2)^9
= (5/4)(512)
= -640
Well, isn't that fascinating! Let's have some fun with math and figure this out!
In a geometric progression (G.P.), each term is found by multiplying the previous term by a constant value called the common ratio. Let's call this common ratio 'r'.
So, given that the third term of the G.P. is 5, we can write the equation: a * r^2 = 5.
And given that the sixth term is -40, we can write: a * r^5 = -40.
Now, let's do some clown calculations. But be prepared, these answers may come with a side of laughter!
Dividing the second equation by the first equation, we get: (a * r^5) / (a * r^2) = -40 / 5.
Simplifying that, we have: r^3 = -8.
So, r = -2. (Remember, we're looking for humor here, not just solutions!)
Now that we have the common ratio, we can find the first term (a) by substituting it into either equation. Let's use the first one: a * (-2)^2 = 5.
That simplifies to a * 4 = 5.
Hence, a = 5/4. (I don't know about you, but I think fractions have a funny bone!)
Now, to find the tenth term, we use the formula: Tₙ = a * r^(n-1).
Substituting in the known values, we get: T₁₀ = (5/4) * (-2)^(10-1).
Laughing a little, we have T₁₀ = (5/4) * (-2)^9.
Still chuckling, we get T₁₀ = (5/4) * (-512).
And finally, bursting with laughter, we find the tenth term of the sequence is: T₁₀ = -640/4.
So, dear friend, the tenth term is -160! (Oh, the hilarity!)
Remember, laughter is the best math teacher! I hope you had a good laugh and found this answer amusing.
To find the 10th term of the geometric progression (G.P.), we need to determine the common ratio (r) of the sequence.
Given that the third term is 5 and the sixth term is -40, we can set up two equations to solve for the common ratio:
1. a * r^2 = 5 --- (Equation 1)
2. a * r^5 = -40 --- (Equation 2)
where 'a' represents the first term.
To eliminate 'a' and solve for 'r', we can divide Equation 2 by Equation 1:
(a * r^5) / (a * r^2) = -40 / 5
r^3 = -8
Now, we take the cube root of both sides of the equation to isolate 'r':
r = -2
Now that we have the common ratio (r = -2), we can find the first term 'a' using Equation 1:
a * r^2 = 5
a * (-2)^2 = 5
4a = 5
a = 5 / 4 = 1.25
Now that we know the first term (a = 1.25) and the common ratio (r = -2), we can find the tenth term (T10) using the formula for the nth term of a geometric progression:
Tn = a * r^(n-1)
T10 = 1.25 * (-2)^(10-1)
T10 = 1.25 * (-2)^9
T10 = 1.25 * (-512)
T10 = -640
Therefore, the tenth term of the sequence is -640.
To find the tenth term of the geometric progression (G.P.), we first need to find the common ratio (r) of the sequence.
In a G.P., each term is found by multiplying the previous term by the common ratio. Let's assume the first term of the G.P. is 'a', and the common ratio is 'r'.
Given:
Third term = 5
Sixth term = -40
We know that the third term is found by multiplying the first term (a) by the common ratio (r) twice:
a * r * r = 5 ----(1)
Similarly, the sixth term is found by multiplying the first term (a) by the common ratio (r) five times:
a * r * r * r * r * r = -40 ----(2)
To find the common ratio (r), we can divide equation (2) by equation (1):
-40 / 5 = r^5 / r^2
Simplifying, we get:
-8 = r^3
Now, we can find the value of 'r' by taking the cube root of both sides:
r = ∛(-8)
r = -2
Now we know the common ratio (r = -2), and we need to find the tenth term of the G.P.
We can use the formula for the nth term of a G.P., which is given by:
tn = a * r^(n-1)
Here, n = 10 (tenth term), r = -2 (common ratio), and we still need to find the value of 'a' (first term).
To find 'a', we can substitute the values of the third term (5) and the common ratio (-2) into equation (1):
a * (-2) * (-2) = 5
4a = 5
a = 5/4
Now we have all the values we need to find the tenth term (t10):
t10 = (5/4) * (-2)^(10-1)
t10 = (5/4) * (-2)^9
Simplifying, we find:
t10 = (5/4) * (-512)
t10 = -640
Therefore, the tenth term of the geometric progression is -640.