In a Geometric progression the sum of its first 4 term is equal 15, and the sum of the next term is equal to 15/16. find its first term and its common ratio
the first term of a geometric progession is 5 and the third term is 20. Find the commonratio of this progession.
To find the first term and common ratio of a geometric progression, you can use the formulas for the sum of a geometric series.
Let's denote the first term as 'a' and the common ratio as 'r'.
1. Finding the Sum of the First 4 Terms:
The sum of the first four terms of a geometric progression is given by the formula: S4 = a(1 - r^4) / (1 - r), where S4 represents the sum of the first 4 terms.
Given that S4 = 15, we have: 15 = a(1 - r^4) / (1 - r).
2. Finding the Sum of the Next Term:
The sum of the next term (5th term) is given by the formula: S5 = a(1 - r^5) / (1 - r).
Given that S5 = 15/16, we have: 15/16 = a(1 - r^5) / (1 - r).
We now have a system of two equations with two variables:
Equation 1: 15 = a(1 - r^4) / (1 - r)
Equation 2: 15/16 = a(1 - r^5) / (1 - r)
Solving this system of equations will give us the values of 'a' and 'r', which represent the first term and common ratio of the geometric progression, respectively.
Let's proceed with solving this system of equations.