If told that the 3rd term of a GP is 36 and the 8th term is 8748. find the first term and the common ration.
start with 4 with a factor of 3
4 12 36 for the 3rd
The n - th term of a geometric progression with initial value a and common ratio r is given by:
an = a * r ^ ( n - 1 )
In this case :
a3 = a * r ^ ( 3 - 1 ) = a * r ^ 2 = 36
a8 = a * r ^ ( 8 - 1 ) = a * r ^ 7 = 8748
So you must solve two equations :
a * r ^ 2 = 36
and
a * r ^ 7 = 8748
a * r ^ 2 = 36 Divide both sides by r ^ 2
a * r ^ 2 / r ^ 2 = 36 / r ^ 2
a = 36 / r ^ 2
a * r ^ 7 = 8748 Divide both sides by r ^ 7
a * r ^ 7 / r ^ 7 = 8748 / r ^ 7
a = 8748 / r ^ 7
a = a
36 / r ^ 2 = 8748 / r ^ 7 Multiply both sides by r ^ 7
36 * r ^ 7 / r ^ 2 = 8748 * r ^ 7 / r ^ 7
36 * r ^ 5 = 8748 Divide both sides by 36
36 * r ^ 5 / 36 = 8748 / 36
r ^ 5 = 243
r = fifth root ( 243 )
r = 3
a = 36 / r ^ 2
a = 36 / 3 ^ 2 = 36 / 9 = 4
OR
a = 8748 / r ^ 7
a = 8748 / 3 ^ 7 = 8748 / 2187 = 4
The first term of a GP:
a = 4
The common ratio:
r = 3
To find the first term and the common ratio of a geometric progression (GP), we can use the given information about the 3rd and 8th terms.
Let's assume the first term of the GP is "a" and the common ratio is "r".
The formula for the nth term of a GP is given by:
Tn = a * r^(n-1)
Given that the 3rd term is 36, we can substitute n = 3 into the formula:
36 = a * r^(3-1) = a * r^2 ---(1)
Similarly, given that the 8th term is 8748, we can substitute n = 8:
8748 = a * r^(8-1) = a * r^7 ---(2)
We now have two equations, (1) and (2), with two unknowns, "a" and "r". We can solve this system of equations simultaneously.
From equation (1), we can express "a" in terms of "r":
a = 36 / r^2 ---(3)
Substituting (3) into equation (2), we have:
8748 = (36 / r^2) * r^7
Simplifying further:
8748 = 36 * r^(7 - 2)
8748 = 36 * r^5
Dividing both sides by 36:
r^5 = 8748 / 36
r^5 = 243
Taking the 5th root of both sides:
r = 3
Now that we have the value of "r" as 3, we can substitute it back into equation (3) to find "a":
a = 36 / (3^2)
a = 36 / 9
a = 4
Therefore, the first term of the GP is 4, and the common ratio is 3