Adding Exponents

[cdwashin]

Hello can someone please explan the following in detail.

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

The answer is 2^9.

Thank you.

[RonPurewal]

ATTENTION: PLEASE POST ALL ANSWER CHOICES WITH PROBLEMS FROM NOW ON.
not only is this the rule around here, but, if you poke around, you'll notice that EVERY other post in this folder does likewise.
from now on, we won't answer posts that don't include all the answer choices.

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the "textbook" way to actually do this problem is pretty obnoxious. since it's pretty low down my personal totem pole when i actually use this problem in private lessons, i'll describe it at the end of this post.

but, first, a lecture of sorts.

DO NOT BE LAZY.
in other words:
you can EASILY just DO THE ARITHMETIC on this problem in the allotted time.
2 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256
you're only adding up nine numbers, only two of which are more than two digits. and the correct answer is 512 = 2^9, which is the next power of 2 after 256.
YOU SHOULD NOT BE GETTING THIS PROBLEM WRONG, BECAUSE YOU CAN SOLVE IT JUST BY DOING ARITHMETIC.

seriously:
don't be lazy.
if a problem has an "ugly" solution - whether that be grinding out a bunch of arithmetic (like this one), making some sort of exhaustive list, or whatever else - then just GET STARTED ON IT RIGHT AWAY, unless you can IMMEDIATELY think of something else to do.

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there are at least three other ways to solve this problem.

1. ESTIMATE:
since the answer choices are so far apart, you can just estimate the sum.
the sum (2 + 2 + 4 + 8 + 16 + 32) + (64 + 128) + 256
is, VERY APPROXIMATELY,
60 + 200 + 250
which is 510.

what's more, once you've estimated 2^8 = 250ish, you have convenient estimates: 2^9 = 500ish and 2^10 = 1000ish. the other three answer choices (which are, if i remember correctly, 2^16, 2^35, and 2^37) are simply beyond the pale.
2^9, which matches the estimate almost exactly, is the only one that's even close.


2. PATTERN RECOGNITION:
there's clearly nothing special about cutting off this pattern at 2^8 - i.e., that's just a random place to stop the sequence - so, add up SMALLER versions of the sequence until you NOTICE A PATTERN.
first term: 2
first two terms: 2 + 2 = 4
first three terms: 2 + 2 + 4 = 8
first four terms: 2 + 2 + 4 + 8 = 16
hey, i see a pattern! it's doubling every time.
you've got 5 more terms to go, so just double your answer 5 more times, giving 512.


3. TEXTBOOK METHOD:
as i've mentioned, this method is a bit obnoxious relative to all the simpler methods above, but here it is anyway.

the first two terms are 2 + 2, which is 2(2), or 2^2.
now add the THIRD term to this: 2^2 + 2^2 = 2(2^2), which is 2^3. (if you don't understand the reason why this is 2^3, write out the first 2 as 2^1, and use your laws of exponents.)
now add the NEXT term: 2^3 + 2^3 = 2(2^3), which is 2^4.
now add the NEXT term: 2^4 + 2^4 = 2(2^4), which is 2^5.
hey, there's a pattern!
if you continue this pattern all the way up until you get to 2^8 (the last term), the total sum will be 2^9.

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note: if the sum went all the way up to, say, 2^100, then, of all the options posted above, only the "textbook method" would be viable.