What type of sequence is the sequence represented by the sum of the terms that has the resul 2^n - 1 ?
A sum of n terms of a general geometric progression which has terms such that consecutive terms have a common ratio is given by a*(r^n - 1) / (r - 1), where a is the first term and r is the common ratio.
Here the sum of the terms is is given as 2^n - 1 which is equal to 1*(2^n - 1)/(2 - 1)
The type of sequence is a geometric progression with first term 1 and the common ratio 2.
The type of sequence represented by the sum of the terms being equal to 2^n - 1 is a geometric progression.
First, we need to determine the general term of the sequence, bn, and then, we'll utter any other term of the progression.
bn = Sn - S(n-1)
bn = (2^n) - 1 - 2^(n-1) + 1
We'll eliminate like terms and we'll factorize by 2^n:
Since we know the general term bn, we'll compute the first 3 consecutive terms, b1,b2,b3.
b1=2^(1-1) = 2^0 = 1
b2=2^(2-1) = 2 = 2*b1
b3=2^(3-1) = 2^2=2*b2
Sn = 2^n - 1 is the sum of the terms of a geometric sequence, whose common ratio is q = 2.