A GP has a first term of a, a common ratio of r and its 6th term is 768. Another GP has a first term of a, a common ratio of 6r and its 3rd term is 3456. Evaluate a and r.
In GP:
an = a ∙ rⁿ⁻¹
First condition:
a6 = 768
a ∙ r⁵ = 768
Second condition:
a3 = 3456
a ∙ ( 6 r)² = 3456
a ∙ 36 r² = 3456
a ∙ r⁵ = 768
:
a ∙ 36 r² = 3456
_____________
a / a ∙ r⁵ / 36 r² = 768 / 3456
1 ∙ r³ / 36 = 384 ∙ 2 / 384 ∙ 9
r³ / 36 = 2 / 9
Multiply both sides by 36
r³ = 2 ∙ 36 / 9
r³ = 72 / 9
r³ = 8
r = ³√ 8
r = 2
First condition:
a6 = 768
a ∙ r⁵ = 768
a ∙ 2⁵ = 768
a ∙ 32 = 768
a = 768 / 32
a = 24
r=2, a=24
To find the values of "a" and "r," we can set up a system of equations using the given information.
Given:
For the first geometric progression (GP):
First term = a
Common ratio = r
Sixth term = 768
For the second geometric progression (GP):
First term = a
Common ratio = 6r
Third term = 3456
We will use these equations to solve for "a" and "r".
First GP:
The formula for the nth term of a geometric progression is given by:
an = a * r^(n - 1)
Using this formula, we have the following equation for the sixth term of the first GP:
768 = a * r^(6 - 1)
768 = a * r^5 ----(Equation 1)
Second GP:
Using the same formula for the nth term, we can set up the equation for the third term of the second GP:
3456 = a * (6r)^(3 - 1)
3456 = a * (6r)^2
3456 = a * 36r^2 ----(Equation 2)
Now, we have a system of equations:
Equation 1: 768 = a * r^5
Equation 2: 3456 = a * 36r^2
We can solve this system of equations to find the values of "a" and "r".
Divide Equation 2 by 36:
96 = a * r^2 ----(Equation 3) (Divided by 36)
Rearrange Equation 3:
r^2 = 96/a ----(Equation 4) (Divided by a)
Substitute the value of r^2 from Equation 4 into Equation 1:
768 = a * (96/a)^5
768 = a * (96^5)/a^5
Cancel out "a" from the numerator and denominator:
768 = 96^5/a^4
Multiply both sides by a^4:
768 * a^4 = 96^5
Now, we can solve for "a" by taking the fourth root of both sides:
a^4 = (96^5)/768
a^4 = 96^3
Take the fourth root of both sides to find "a":
a = ∛(96^3)
Calculating this, we find:
a ≈ 96
Now, substitute the value of "a" back into Equation 1 to solve for "r":
768 = (96) * r^5
Divide both sides by 96:
8 = r^5
Now, take the fifth root of both sides to find "r":
r = ∛8
Simplifying, we have:
r = 2
Therefore, the values of "a" and "r" are:
a = 96
r = 2
To evaluate the values of a and r in the given geometric progressions (GPs), we can use the formulas for the nth term of a GP:
For the first GP, with first term a and common ratio r, the nth term is given by:
nth term = a * r^(n-1)
For the second GP, with first term a and common ratio 6r, the nth term is given by:
nth term = a * (6r)^(n-1)
We are given that the 6th term of the first GP is 768, so we can substitute the values into the formula:
768 = a * r^(6-1) = a * r^5. ----(equation 1)
We are also given that the 3rd term of the second GP is 3456, so we can substitute the values into the formula:
3456 = a * (6r)^(3-1) = a * (6r)^2 = a * 36r^2. ----(equation 2)
Now, we have a system of two equations (equation 1 and equation 2) that we can solve simultaneously to find the values of a and r.
From equation 2, we can rewrite it as:
a = 3456 / (36r^2) = 96 / r^2. ----(equation 3)
Substitute the expression for a from equation 3 into equation 1:
768 = (96 / r^2) * r^5 = 96 * r^(5-2) = 96 * r^3.
Now, we can solve this equation for r by dividing both sides by 96:
768 / 96 = r^3.
Simplifying, we get:
8 = r^3.
Taking the cube root of both sides, we find that:
r = 2.
Substitute this value of r back into equation 1 to calculate the value of a:
768 = a * (2)^5 = 32a.
Solving for a, we divide both sides of the equation by 32:
768 / 32 = a,
a = 24.
Therefore, the values of a and r are a = 24 and r = 2, respectively.
Oh, I see. It's all about finding the values of "a" and "r" for these geometric progressions (GP). Let's dive right in! 🤡
For the first GP, we know that the 6th term is 768. Using the formula for the nth term of a GP, we can write it as:
a * r^5 = 768
Similarly, for the second GP, the 3rd term is 3456. Applying the same formula:
a * (6r)^2 = 3456
Now, let's solve these equations using some algebraic magic. ✨
From the first equation, we can rewrite it as:
a = 768 / r^5
Plugging this into the second equation, we get:
(768 / r^5) * (6r)^2 = 3456
Simplifying this further:
(768 / r^5) * 36r^2 = 3456
Multiplying both sides by r^5 to get rid of the fraction:
768 * 36r^2 = 3456 * r^5
Simplifying again:
27648r^2 = 3456r^5
Dividing both sides by r^2 (assuming r is non-zero):
27648 = 3456r^3
Finally, dividing both sides by 3456:
8 = r^3
Ahoy! We have found our first clue! "r" is the cube root of 8, which is simply 2. 🎉
Now, let's find the value of "a" by substituting "r" into one of our earlier equations:
a = 768 / (2^5)
After evaluating this, we find that "a" equals 12.
Hence, the values of "a" and "r" are a = 12 and r = 2.
Hope that brings a smile to your face! 😄