2/(1-r) = 3
Sn = 2(1-r^n)/(1-r)
Sn = 2(1-r^n)/(1-r)
1. Common Ratio (r) Formula:
The common ratio (r) can be found using the formula:
r = Sum / First term
2. Sum of an Infinite G.P. Formula:
The sum of an infinite G.P. can be found using the formula:
S = First term / (1 - r)
Let's calculate the common ratio (r) and the sum of the series (S) using the given information:
First term (a) = 2
Sum of the series (S) = 3
1. Common Ratio (r) Calculation:
r = S / a
r = 3 / 2
r = 1.5
Therefore, the common ratio (r) is 1.5.
2. Sum of the Series (S) Calculation:
S = a / (1 - r)
S = 2 / (1 - 1.5)
The sum of an infinite G.P. diverges if |r| ≥ 1. In this case, |r| = 1.5, which is greater than 1. Therefore, the sum of the infinite G.P. diverges.
As for finding the sum up to a certain number of terms (n), we can use the formula for the sum of n terms of a G.P.:
Sum of n terms = a * (1 - r^n) / (1 - r)
However, since the infinite G.P. diverges, the sum of a finite number of terms will not provide a meaningful result.
Hence, the common ratio is 1.5, and the sum of the infinite G.P. is divergent.