2+2+2^2+2^3+2^4+2^5+2^6+2^7+2^8=?

VJesus12 wrote:2+2+2^2+2^3+2^4+2^5+2^6+2^7+2^8=?

A. 2^9
B. 2^10
C. 2^16
D. 2^35
E. 2^37

[spoiler]OA=A[/spoiler]

Source: GMAT Prep

Let's look for a pattern...

We want: 2 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 = ?
2 + 2 + 2^2 = 4 + 4 = 8 = 2^3
So, 2 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 = 2^3 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8

2^3 + 2^3 = 2(2^3) = 2^4
So, 2^3 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 = 2^4 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8

2^4 + 2^4 = 2(2^4) = 2^5
So, 2^4 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 = 2^5 + 2^5 + 2^6 + 2^7 + 2^8

Continuing the pattern, we get: 2 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 = 2^9

Answer: A

Cheers,
Brent

VJesus12 wrote:2+2+2^2+2^3+2^4+2^5+2^6+2^7+2^8=?

A. 2^9
B. 2^10
C. 2^16
D. 2^35
E. 2^37

[spoiler]OA=A[/spoiler]

Source: GMAT Prep

We can use the formula:

2^0 + 2^1 + 2^2 + … + 2^n = 2^(n + 1) - 1

Therefore,

2 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8

= 1 + 2^0 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8

= 1 + (2^9 - 1)

= 2^9

Alternate Solution:

If you don’t recall the formula given in the solution above, use arithmetic, as follows:

The first two terms are 2 + 2 + 4. Then add [2^2=4] to get 8, then add [2^3=8] to get 16, then add [2^4 =16] to get 32, then add [2^5=32] to get 64, and then add [2^6=64] to get 128. One more addition is 128 + [2^7=128] = 256. We see the pattern that has evolved, so we know that 256 = 2^8.

So the final addition is 2^8 + 2^8. Pulling out the common factor 2^8, we have:

2^8(1 + 1) = 2^8 x 2 = 2^9.

Answer: A