the third terms of GP is 10 and the 6 term is 80 find the sun of the first 7 terms
To find the sum of the first 7 terms of a geometric progression (GP), we need to first find the common ratio (r) of the sequence.
In a GP, each term is obtained by multiplying the preceding term by a constant ratio. Let's denote the first term as 'a' and the common ratio as 'r'.
Given that the third term is 10, we can express this as:
a * r * r = 10 (Equation 1)
Similarly, the 6th term is 80, so we have:
a * r * r * r * r * r = 80 (Equation 2)
Dividing Equation 2 by Equation 1, we get:
(r * r * r * r * r) / (r * r) = 80 / 10
r * r * r * r = 8
Taking the fourth root of both sides gives:
r = ∛8 = 2
Now that we have the common ratio (r = 2), we can find the first term (a).
Substituting the value of r in Equation 1, we get:
a * 2 * 2 = 10
4a = 10
a = 10 / 4 = 2.5
Therefore, the first term (a) is 2.5 and the common ratio (r) is 2.
To find the sum of the first 7 terms, we can use the formula for the sum of a GP:
Sum = a * (r^n - 1) / (r - 1)
where:
a = first term = 2.5
r = common ratio = 2 (as calculated earlier)
n = number of terms = 7
Substituting these values into the formula:
Sum = 2.5 * (2^7 - 1) / (2 - 1)
= 2.5 * (128 - 1) / 1
= 2.5 * 127
= 317.5
Therefore, the sum of the first 7 terms of the geometric progression is 317.5.