2 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8
= 1 + (1 + 2 + 2^2 + 2^3 + ... + 2^8)
= 1 + (2^0 + 2^1 + 2^2 + 2^3 + ... + 2^8)
the portion
(2^0 + 2^1 + 2^2 + 2^3 + ... + 2^8)
take the 1st 2 terms
2^0 + 2^1 = 1 + 2 = 3
it's identical to 2^2 - 1
therefore,
2^0 + 2^1 + ... + 2^8
= 2^9 - 1
If we add the 1 back, we have
2^9-1 + 1 = 2^9
pattern
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Here's another way to look at it.ST wrote:2 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 =
1) 2^9
2) 2^10
3) 2^16
4) 2^35
5) 2^37
answer is 1
anyone know why answer is not 5?
Please note that 2^7 is half as large as 2^8 and, in general, 2^n is half as large as 2^(n+1)
Let's say that I have a lot of gold. I have 2^9 kg.
Today I give you half of all my gold. In other words, I give you 2^8 (which leaves me with 2^8 kg).
Tomorrow I give you half of my remaining gold. I give you 2^7 (which leaves me with 2^7 kg).
Each day I give you half of my remaining gold.
When I get to the eighth day, I give you 2 kg of gold (which leaves me with 2 kg)
Then, on the last day, I give you my last 2 kg of gold.
I have now given you all of my gold (all 2^9 kg)
So, we can see that 2^8 + 2^7 + 2^6 + 2^5 + 2^4 + 2^3 + 2^2 + 2 + 2 + 2^9
Brent Hanneson - Creator of GMATPrepNow.com
I think this can be solved in following way...
geometric progression formula
Sum= a1 (r^n-1)
--------------
(r-1)
in given equation a1=2; r=2
gives 2(2^8-1)
2+ -----------
(2-1)
solve you get 2^9 as your answer
geometric progression formula
Sum= a1 (r^n-1)
--------------
(r-1)
in given equation a1=2; r=2
gives 2(2^8-1)
2+ -----------
(2-1)
solve you get 2^9 as your answer
