the 8th terms of a geometry progression is 640. if the first term is 5, find the common ratio and the 10th terms?
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640 = 5r^7
128 = r^7
r = 128^(1/7) = 2 of course
so what is 5 * 2^9 ?
To find the common ratio of a geometric progression, we can use the formula:
tn = a * r^(n - 1)
where:
tn = the nth term
a = the first term
r = the common ratio
n = the term number
We are given the value of the 8th term (tn) as 640, and the first term (a) as 5. So we can plug these values into the formula:
640 = 5 * r^(8 - 1)
Simplifying this equation, we have:
640 = 5 * r^7
To find the value of r, we can divide both sides of the equation by 5:
(r^7) = (640 / 5)
(r^7) = 128
Now, to find the value of r, we need to take the 7th root of both sides:
r = (128)^(1/7)
Calculating this using a calculator or by hand, we get:
r ≈ 2
So, the common ratio is 2.
Now, to find the 10th term of the geometric progression, we can use the same formula:
tn = a * r^(n - 1)
Plugging in the values we know:
t10 = 5 * 2^(10 - 1)
Simplifying this:
t10 = 5 * 2^9
t10 = 5 * 512
t10 = 2560
Therefore, the 10th term of the geometric progression is 2560.
To find the common ratio, we'll use the formula for the nth term of a geometric progression:
nth term = a * r^(n-1)
where a is the first term and r is the common ratio.
Given that the 8th term is 640 and the first term is 5, we can plug these values into the formula:
640 = 5 * r^(8-1)
Simplifying this equation:
640 = 5 * r^7
Dividing both sides by 5:
128 = r^7
To solve for r, we'll take the 7th root of both sides:
r = 7√128
Calculating the 7th root of 128:
r ≈ 2.2974
So, the common ratio (r) is approximately 2.2974.
To find the 10th term, we can substitute the values into the formula:
nth term = 5 * r^(n-1)
10th term = 5 * (2.2974)^(10-1)
Calculating this expression:
10th term ≈ 173.7654
Therefore, the 10th term is approximately 173.7654.