Decide if the expression 4^n -1 is the sum of terms of an a.p. or g.p.?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll determine the general term bn of the sum of the terms,whose expression is 4^n -  1, and then, we'll utter any other term of the progression.

From enunciation:

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

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

bn=(4^n)-1-[b1+b2+b3+...+b(n-1)]

But [b1+b2+b3+...+b(n-1)]=S(n-1)=[4^(n-1)]-1

bn=(4^n) - 1 - 4^(n-1) + 1

We'll eliminate like terms and we'll factorize by 4^n:

bn=4^n(1-1/4)=4^n*3/4=3*4^(n-1)

Since we know the general term bn, we'll compute the first 3 consecutive terms, b1,b2,b3, considering that they are the terms of a geometric progression.

b1=3*4^0

b2=3*4^(2-1)=3*4

b3=3*4^(3-1)=3*4^2

Following the rule of a geometric sequence, we'll verify if

b2=sqrt (b1*b3)

3*4= sqrt(3*1*3*16)

3*4=3*4

Sn = 4^n - 1 is the sum of the terms of a geometrical progression.

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