\(2^{(22)} + 2^{(23)} + 2^{(24)} + ... + 2^{(43)} + 2^{(44)} =\)


A \(2^{22}(2^{(23)}-1)\)

B \(2^{22}(2^{(24)}-1)\)

C \(2^{22}(2^{(25)}-1)\)

D \(2^{23}(2^{(21)}-1)\)

E \(2^{23}(2^{(22)}-1)\)

Hey guys,

GREAT question - thanks for sharing this! This one is a terrific platform for what I call the Three Guiding Principles for Exponents:

1) Find common bases (which usually means you have to factor out bases into prime factors)

2) Multiply (which means that you have to factor out addition/subtraction terms to turn them into multiplication)

3) Find patterns (usually using small numbers to predict how patterns will look with the larger numbers you're given)


Here, I'll use Guiding Principles 2 and 3 (because all the bases are already 2):

All those terms - 2^22 through 2^44 - are going to be too much to carry around, so I'll use a smaller set to try to establish a pattern. Let's say it was just:

2^22 + 2^23 + 2^24 + 2^25

Then we'd need to use Guiding Principle #2 to turn this addition into multiplication, by factoring out the common 2^22 from all terms:

2^22 (1 + 2^1 + 2^2 + 2^3)

2^22 (1 + 2 + 4 + 8)

2^22 (15)

Looking at the answer choices we know we want to have 2^exponent - 1 in there, and we can make the 15 here 16-1, or 2^4 - 1, so we have:

2^22 (2^4 - 1) ----> Started with 4 terms (2^22 through 2^25) and ended up with the 2^22 multiplied by 2^4 - 1


Let's try again, but with five terms to start to see how the pattern goes:

2^22 + 2^23 + 2^24 + 2^25 + 2^26

Factor to:

2^22 (1 + 2 + 2^2 + 2^3 + 2^4)

2^22 (1 + 2 + 4 + 8 + 16)

2^22 (31)

And if we put it in that form as the answer choices we get:

2^22 (2^5 - 1) ----> starting with 5 terms, we end up with 2^22 multiplied by 2^5 - 1


Looking at the pattern, it should become fairly clear that we'll always end up with 2^22 (what we factor) multiplied by 2^an exponent equal to the number of terms we started with, minus 1. So the number of terms that we start with will equal the exponent in 2^x - 1.

Since we'll have 2^22 through 2^44, we have 44-22 + 1 terms (it's an inclusive set), so 23 terms total, and we'll end up with:

2^22 * (2^23 - 1)

And that's answer choice A.
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Dude is that geometric progression formula correct?

Cheers
J

How about this approach -

A = 2^(22) + 2^(23) + 2^(24) + ... + 2^(43) + 2^(44) ... 1
2A = 2^(23) + 2^(24) + ... + 2^(43) + 2^(44) + 2^45 ... 2

2 - 1 gives
A = 2^45 - 2^22 = 2^22 ( 2^23 - 1)

Let’s begin by factoring out 2^22, which is common to each of the terms:

2^22 + 2^23 + … + 2^44 = 2^22(1 + 2 + 2^2 + … + 2^22)

To find an expression for 1 + 2 + 2^2 + … + 2^22, we will use the formula

a^n - 1 = (a - 1)(a^(n - 1) + a^(n - 2) + … + a + 1).

If we let a = 2 and n = 23, we obtain

2^23 - 1 = (2 - 1)(2^22 + 2^21 + … + 2 + 1).

Thus:

2^22 + 2^23 + … + 2^44 = 2^22(1 + 2 + 2^2 + … + 2^22) = 2^22(2^23 - 1)

Answer: A
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