The three term of a GP is 20 and 8 term is 640 find the common ratio. The first term and 10 term of a GP
I will assume you mean:
the third term of a GP is 20 and the 8th term is 640, or else you sentence makes no sense
ar^2 = 20
ar^7 = 640
divide them:
r^5 = 32
r = 2
now you can find a
and then ar^9, which is the 10th term
pls I really don't understand how you solve for this equation r^5 = 32 to derive at 2 as your answer 🤔🤔
To find the common ratio (r) of a geometric progression (GP), we can use the formula:
náµ—Ê° term = a * r^(n-1)
Given that the third term is 20, we can write:
20 = a * r^(3-1)
20 = a * r² ----(1)
Similarly, given that the eighth term is 640, we have:
640 = a * r^(8-1)
640 = a * r^7 ----(2)
Now, let's solve equations (1) and (2) simultaneously to find the common ratio (r):
Dividing equation (2) by equation (1):
(640 / 20) = (a * r^7) / (a * r²)
32 = r^5
Taking the fifth root of both sides:
∛∛∛∛∛32 = r
Therefore, the common ratio (r) is 2.
Now let's find the first term (a) and the 10th term of the GP.
From equation (1) we have:
20 = a * 2²
20 = 4a
Dividing both sides by 4:
a = 20 / 4
a = 5
To find the 10th term, we substitute n = 10 into the formula:
10áµ—Ê° term = 5 * 2^(10-1)
10áµ—Ê° term = 5 * 2^9
10áµ—Ê° term = 5 * 512
10áµ—Ê° term = 2560
Therefore, the first term of the geometric progression is 5 and the 10th term is 2560.
To find the common ratio of a geometric progression (GP), we can use the formula:
a_n = a * r^(n-1)
Where:
- a_n is the n-th term of the GP
- a is the first term of the GP
- r is the common ratio of the GP
- n is the position of the term in the GP
Given that the third term is 20, we can write:
20 = a * r^(3-1)
20 = a * r^2
Similarly, given that the eighth term is 640, we have:
640 = a * r^(8-1)
640 = a * r^7
To find the common ratio, we can divide the second equation by the first equation, like this:
(640 / 20) = (a * r^7) / (a * r^2)
32 = r^(7-2)
32 = r^5
Taking the fifth root of both sides, we have:
r = ∛(32)
r = 2
Therefore, the common ratio is 2.
To find the first term and the tenth term of the GP, we can use the same formula:
a_n = a * r^(n-1)
For the first term (n=1):
a_1 = a * r^(1-1)
a_1 = a * r^0
a_1 = a
So, the first term (a_1) of the GP is equal to 'a'.
For the tenth term (n=10):
a_10 = a * r^(10-1)
a_10 = a * r^9
Now that we know the value of the common ratio (r=2), we can calculate the first and tenth terms of the GP.