The sum of n terms of a series is a.2n-b, find its nth term. are the terms of this series in G.P. ?
okay okay
time for some of your work, please
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To find the nth term of a series given the sum of n terms, you need to use the formula for the sum of an arithmetic series.
The sum of the first n terms of an arithmetic series is given by the formula Sn = (n/2)(a + L), where Sn is the sum of the n terms, a is the first term, and L is the last term.
In this case, the sum of n terms is given as a.2n - b, so we can equate this to Sn and solve for L:
a.2n - b = (n/2)(a + L)
Now, let's rearrange the equation to isolate L:
a.2n - b = (n/2)(a + L)
2.a.2n - 2b = n(a + L)
4an - 2b = na + nL
nL = 4an - na - 2b
nL = n(4a - a) - 2b
nL = 3na - 2b
L = (3na - 2b) / n
So the nth term, which is L, is equal to (3na - 2b) / n.
Now, to determine if the terms of this series are in a geometric progression (G.P.), we need to check if there is a common ratio between consecutive terms. If there is a common ratio, it means the terms are in a G.P.
To check for a common ratio, we can take the ratio of any two consecutive terms. Let's consider the ratio between the (n+1)th term and the nth term:
R = ((3(n+1)a - 2b) / (n+1)) / ((3na - 2b) / n)
= (3na + 3a - 2b) / (n+1). (3na - 2b) / n
= (3na + 3a - 2b) * (n / (n+1)) / (3na - 2b)
As you can see, the ratio R is not a constant value. Therefore, the terms of this series are not in a geometric progression (G.P.).