The sum of n terms of a series is a.2^n-b, find its n th term. Are the terms of this series in g.p.?
sir this question answer (a2^n-1)
To find the nth term of a series, we need to understand the pattern or rule governing the series. In this case, we are given that the sum of n terms of the series is given by the expression a * 2^n - b.
To find the nth term, we can subtract the sum of the first n-1 terms from the sum of the first n terms. This will give us the value of the nth term alone.
So, let's first find the sum of the first n-1 terms:
Sum of the first n-1 terms = a * 2^(n-1) - b
Now, let's find the sum of the first n terms:
Sum of the first n terms = a * 2^n - b
To find the nth term, we subtract the sum of the first n-1 terms from the sum of the first n terms:
nth term = (Sum of the first n terms) - (Sum of the first n-1 terms)
= (a * 2^n - b) - (a * 2^(n-1) - b)
= a * 2^n - b - a * 2^n-1 + b
= a * 2^n - a * 2^n-1
= a * (2^n - 2^n-1)
= a * 2^n * (1 - 1/2)
= a * 2^n * (1/2)
= a * 2^(n-1)
Therefore, the nth term of the series is given by a * 2^(n-1).
Now, let's check if the terms of this series are in a geometric progression (g.p.). In a g.p., each term is obtained by multiplying the previous term by a fixed ratio. Let's check if this holds true for the given series.
If we compare the nth term with the (n-1)th term:
nth term = a * 2^(n-1)
(n-1)th term = a * 2^((n-1)-1) = a * 2^(n-2)
To check if the terms are in g.p., we need to see if we can obtain the nth term by multiplying the (n-1)th term by a fixed ratio.
Ratio = nth term / (n-1)th term = a * 2^(n-1) / (a * 2^(n-2))
= (2^(n-1)) / (2^(n-2))
= 2
Since the ratio between consecutive terms is a constant (2), we can conclude that the terms of this series form a geometric progression (g.p.).
So, the nth term of the series is a * 2^(n-1), and the terms of this series are in a geometric progression (g.p.).