The Product of three Consecutive terms of a G.P. is -64 and the first term is four times the third. find the number.
3 numbers in gp when product is given is: a/r , a , ar .
Now ,
a/r ×a×ar= -64
=a^3=-64
=a= -4
Now it is given that first number is 4 times the third number so;
a/r=4(ar)
-4/r=4(-4r)
-1/r=-4r
1/4=r^2
r=+or-1/2
So we have a =-4 and r=+1/2 and -1/2 .
So we get no. as : a/r,a,ar....as
-8,-4,-2 or 8,-4,2
Ah, a geometric progression problem! Let me put on my clown shoes and try to solve it with a touch of humor!
Let's call the first term "x" and the common ratio "r". Since we're given that the first term is four times the third, we can say that the third term is "x/4".
Now, let's use the formula for the product of three consecutive terms in a G.P.: P = xyz.
In this case, P is given as -64, so we have -64 = x * (x/4) * (x*r). Simplifying, we get -64 = (x^3 * r) / 4.
Since my comedy skills are better than my math skills, let's use some brute force and guess some values, shall we? Let's try r = 2, just for laughs.
Substituting r = 2 into our equation, we have -64 = (x^3 * 2) / 4. Simplifying, we get -128 = x^3 / 2. Hmm, that doesn't seem right. Let's change our approach.
Okay, here's a clown-approved solution! The trick is to realize that since our answer is -64, which is negative, it means that there must be an odd number of negative terms in our G.P.
Let me explain why. In a positive G.P., multiplying positive numbers will always result in a positive product. But if we introduce a negative number into the mix, the product will become negative. And since we're dealing with three terms, there must be an odd number of negatives to get a negative product.
So, let's assume the first term is positive, making the second term negative, and the third term positive again. Therefore, we can say that x = 4, (x/4) = -1, and r = 1, to satisfy our conditions.
Substituting these values into the formula, we have -64 = 4 * (-1) * 4. Simplifying, -64 = -64. Hooray! The universe hasn't collapsed on itself! So, the number we're looking for is 4.
Remember, math can be a juggling act sometimes, but finding the right answer is always worth it!
Let's assume the three terms of the geometric progression (G.P.) to be a, ar, and ar^2. Here, 'a' is the first term, 'r' is the common ratio, and 'ar^2' is the third term.
We are given that the product of the three consecutive terms is -64. Therefore, we can write the equation as:
a * ar * ar^2 = -64
Now, we know that the first term (a) is four times the third term (ar^2). Mathematically, this can be expressed as:
a = 4 * ar^2
Substituting the value of 'a', the equation becomes:
4 * ar^2 * ar * ar^2 = -64
Simplifying the equation, we get:
4 * ar^4 = -64
Dividing both sides of the equation by 4, we have:
ar^4 = -16
Taking the fourth root of both sides, we get:
r = -√2
Now, substituting the value of 'r' in the equation a = 4 * ar^2, we have:
a = 4 * a * (-√2)^2
a = 4 * a * 2
On further simplification:
a = 8a
Dividing by 'a' on both sides:
1 = 8
Since this equation is not true, it means that there is no value of 'a' that satisfies the given conditions.
Hence, there is no solution or number that fits the given conditions.
To solve this problem, we need to use the given information to set up an equation and then solve for the unknown number.
Let's represent the first term of the GP as 'a', the common ratio as 'r', and the third term as 'ar^2'.
From the given information, we can create two equations:
Equation 1: The product of the three consecutive terms is -64
a * ar * ar^2 = -64
Equation 2: The first term is four times the third term
a = 4(ar^2)
Now, let's solve these equations step by step:
1. Substituting the value of 'a' from Equation 2 into Equation 1:
4(ar^2) * ar * ar^2 = -64
2. Simplifying Equation 1:
4ar^3 * ar^4 = -64
4ar^7 = -64
3. Dividing both sides by 4:
ar^7 = -16
4. Taking the 7th root of both sides to isolate 'r':
r = (-16)^(1/7)
5. Finding the value of 'a' using Equation 2:
a = 4(ar^2)
a = 4[(4ar^2)r^2]
a = 16ar^4
6. Substituting the value of 'r' from step 4 into Equation 5:
a = 16a[(-16)^(1/7)]^4
7. Simplifying Equation 6:
a = 16a * (-16)^(4/7)
Now, we need to solve for the value of 'a'.
At this point, it becomes apparent that the given information may not have a unique solution. This is because there may not exist a real number that satisfies both equations.
r = 1/2
try that to get started.