What type of sequence is the sequence represented by the sum of the terms that has the resul 2^n - 1 ?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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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.

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giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

First, we need to determine the general term of the sequence, bn, and then, we'll utter any other term of the progression.

From enunciation:

Sn=b1+b2+b3+...+bn

(2^n)-1=b1+b2+b3+...+bn

bn=(2^n)-1-[b1+b2+b3+...+b(n-1)]

But [b1+b2+b3+...+b(n-1)]=S(n-1)=[2^(n-1)]-1

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:

bn=2^n(1-1/2)=2^n*1/2=2^(n-1)

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.

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