A g.p has a first term of "a", a common ratio of "r" and its 6th term is 768. Another G.P has a first term of "a", common ratio of 6r and its 3rd term is 3456. Evaluate "a and r".
1st GP
"6th term is 768" ---> ar^5 = 768
2nd GP, a = a, r = 6r
3rd term = 3456
a(6r)^2 = 3456
36a r^2 = 3456
36a r^2 / (ar^5) = 3456/768
36/r^3 = 9/2
9r^3 = 72
r^3 = 8
r = 2
back in ar^5 = 768
a(2^5) = 768
a = 24
Tn = a r^(n-1)
T6 = a r^5 = 768
T'n' = a (6r)^(n'-1)
T'3 = a (6r)^2 = 36 a r^2 = 3456
a r^5 /36 a r^2 = 768/3456
r^3 = 8
r = 2
your turn
To find the values of "a" and "r," we can use the formulas for the nth term of a geometric progression (G.P).
For the first G.P, we are given the 6th term as 768. We can use the formula for the nth term of a G.P:
an = a * r^(n-1)
The 6th term of the first G.P is 768, so we have:
768 = a * r^(6-1)
768 = a * r^5
For the second G.P, we are given the 3rd term as 3456. Using the same formula, we have:
3456 = a * (6r)^(3-1)
3456 = a * (6r)^2
3456 = a * 36r^2
Now we have a system of two equations:
1) 768 = a * r^5
2) 3456 = a * 36r^2
To solve this system, we can divide equation 2 by equation 1:
3456 / 768 = (a * 36r^2) / (a * r^5)
4.5 = 36r^2 / r^5
4.5 = 36 / r^3
(4.5 * r^3) = 36
Now we can solve this equation for "r":
r^3 = 36 / 4.5
r^3 = 8
r = ∛(8)
r = 2
Substituting the value of "r" into equation 1 to find "a":
768 = a * (2^5)
768 = a * 32
a = 768 / 32
a = 24
Therefore, the values of "a" and "r" are a = 24 and r = 2, respectively.
To find the values of "a" and "r" in both geometric progressions (G.P.), we will use the formulas for finding the nth term of a G.P.
In the first G.P., the first term is "a" and the common ratio is "r". We're given that the 6th term is 768. The formula for the nth term of a G.P. is given by:
An = a * r^(n-1),
where An represents the nth term. Substituting the values, we have A6 = a * r^5 = 768.
Similarly, in the second G.P., the first term is still "a", but the common ratio is now 6r. The 3rd term is 3456. Applying the formula again, we have A3 = a * (6r)^(3-1) = a * (6r)^2 = 3456.
Now we have a system of equations:
Equation 1: a * r^5 = 768 ----(1)
Equation 2: a * (6r)^2 = 3456.
We can solve this system of equations to find the values of "a" and "r." Let's solve it step by step:
From Equation 1, we can express "r" in terms of "a" as:
r = (768 / a)^(1/5).
Substituting this into Equation 2, we have:
a * (6 * (768/a)^(1/5))^2 = 3456.
Simplifying further, we get:
6^2 * 768^(2/5) = 3456 / a.
Rearranging, we find:
a = 3456 / (6^2 * 768^(2/5)).
Now that we have the value of "a," we can substitute it back into Equation 1 to find "r":
r = (768 / a)^(1/5).
Performing these calculations will give you the values of "a" and "r" in both geometric progressions.