First term and the sum of an infinite G.P. are 2 and 3 respectively. find the common ratio and sum upto n term.
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First term and the sum of an infinite GP are 2 and 3 respectively find the common ratio and sum upto n term
To find the common ratio and sum of an infinite Geometric Progression (G.P.), given the first term and the sum of the infinite series, you can use the following formulas:
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.
2/(1-r) = 3
Sn = 2(1-r^n)/(1-r)