If the third term of a G.p is four times it's fifth term. Find the possible values of the common ratios
5th / 3rd = 1/4 = r^2
r = √(1/4) = ±1/2
+1/2 or -1/2
Oh, the possible values of the common ratio in this case? Well, let me put on my clown nose and calculate that for you!
Let's say the common ratio of our geometric progression is "x". According to the given information, the third term would be "x^2" (since we multiply the previous term, which is "x", by "x" again). And the fifth term would be "x^4".
Now, we know that the third term is four times the fifth term. So we can set up an equation:
x^2 = 4 * x^4
Solving this equation, we find that x^4 = 1/4. Taking the fourth root of both sides, we get:
x = ±(1/2)
So, the possible values for the common ratio "x" are 1/2 and -1/2. I hope that puts a smile on your face!
Let's assume the first term of the geometric progression (G.P.) is "a" and the common ratio is "r".
The third term of the G.P. can be represented as ar^2, and the fifth term can be represented as ar^4.
Given that the third term is four times the fifth term, we have:
ar^2 = 4(ar^4)
Let's simplify the equation:
ar^2 = 4ar^4
Dividing both sides by "a" (assuming a≠0):
r^2 = 4r^4
Rearranging the equation:
4r^4 - r^2 = 0
Factoring out r^2:
r^2(4r^2 - 1) = 0
Now, we have two separate equations:
r^2 = 0 or 4r^2 - 1 = 0
For r^2 = 0, the possible values for r are:
r = √0 = 0
For 4r^2 - 1 = 0, the possible values for r are:
4r^2 = 1
Dividing both sides by 4:
r^2 = 1/4
Taking the square root of both sides:
r = ±√(1/4)
Simplifying:
r = ±1/2
Therefore, the possible values for the common ratio (r) are 0, 1/2, and -1/2.
To find the common ratio of a geometric progression (G.P.), we need to analyze the relationship between the terms.
Let's assume the common ratio of the G.P. is denoted by r.
The G.P. is defined as a sequence where each term after the first is found by multiplying the previous term by a constant (the common ratio, r).
Let's denote the third term as T3 and the fifth term as T5.
According to the given information, the third term (T3) is four times the fifth term (T5). We can express this mathematically as:
T3 = 4T5
Now, let's substitute the terms using the general formula for a G.P:
T3 = ar^2 (since a is the first term, and T3 = a * r^2)
T5 = ar^4 (since a is the first term, and T5 = a * r^4)
By substituting these expressions into the equation T3 = 4T5, we get:
ar^2 = 4(ar^4)
Now, divide both sides by a to eliminate it:
r^2 = 4r^4
Next, rearrange the equation to bring all the terms on one side in the form of a quadratic equation:
4r^4 - r^2 = 0
Factor out common terms:
r^2(4r^2 - 1) = 0
Now, set each factor equal to zero:
r^2 = 0 or 4r^2 - 1 = 0
For r^2 = 0, we have r = 0 (since the square of any number is equal to zero only if the number itself is zero).
For 4r^2 - 1 = 0, add 1 to both sides:
4r^2 = 1
Divide both sides by 4:
r^2 = 1/4
Taking the square root of both sides, we get:
r = ±√(1/4)
Simplifying the square root, we obtain two possible values for r:
r = ±1/2
Therefore, the possible values for the common ratio (r) of the geometric progression are ±1/2 or -1/2 and 1/2.