Please solve this exercise:
\[1 + 2 + 2^2 + 2^3 + 2^4 + \ldots + 2^{2011}\]
Can it be solved by using a geometric progression?
Answer (2)
1 + 2 + 2 2 + 2 3 + 2 4 + ... + 2 2011 a 1 = 1 ; a 2 = 1 ⋅ 2 = 2 ; a 3 = 2 ⋅ 2 = 2 2 ; a 4 = 2 2 ⋅ 2 = 2 3 ⋮ a 2012 = 2 2010 ⋅ 2 = 2 2011
T h e s u m o f a t er m s o f g eo m e t r i c p ro g ress i o n : S n = 1 − r a 1 ( 1 − r n ) a 1 = 1 ; r = 2 s u b t i t u t e : S 2012 = 1 − 2 1 ( 1 − 2 2012 ) = − 1 1 − 2 2012 = 2 2012 − 1 O n l y t ha t ... ( 606 d i g i t s , i f yo u w an t h o w l e n g t h t hi s n u mb er )
The series 1 + 2 + 2 2 + 2 3 + ... + 2 2011 is a geometric progression with first term 1 and common ratio 2, totaling 2012 terms. The sum can be calculated using the formula for geometric series, giving the result 2 2012 − 1 . This formula effectively provides the sum of all powers of 2 from 0 to 2011.
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