# Adding 2 Powers of 1 Half

## When adding 2 powers of 1 half(½), the most likely numerator that the sum will have is 3!

## (½)^{x} + (½)^{x+1} = 3/(2^{x+1})

### Or the formula can be written like this:

## 2^{-x} + 2^{-x-1} = 3 × 2^{-x-1}

__Examples:__

(½)^{1} + (½)^{2} = 1/2 + 1/4 = 3/4

(½)^{2} + (½)^{3} = 1/4 + 1/8 = 3/8

(½)^{3} + (½)^{4} = 1/8 + 1/16 = 3/16

(½)^{4} + (½)^{5} = 1/16 + 1/32 = 3/32

(½)^{5} + (½)^{6} = 1/32 + 1/64 = 3/64

Even if *x* = **0,** the equation remains true!

(½)^{0} + (½)^{1} = 1 + 1/2 = 3/2

You already know that raising any non-zero number to the power of **zero** gives you **1,** right? The equation will still be technically true if *x* is not an integer, but the sum will not be a rational number! All rational numbers are a quotient of 2 integers!

(½)^{½} + (½)^{3/2} = the square root of 1/8 + the square root of 1/2 = 1.060660172...

The irrational number in the counterexample is equal to **3 ÷ the square root of 8.** Normally in fractions, people write numerators & denominators as integers; although sometimes in advanced algebra & calculus, mathematicians write variables or famous irrational numbers like **pi(π)** as numerators or denominators! That is done to be more exact! Besides, it'll take forever to write or print all those digits after the decimal point; since there's no digit pattern in irrational numbers, you never know which digit to put next! __Most real numbers are irrational!__

### Below is how the function looks on a graph:

Notice that I printed the function as *(½)*^{x} + (½)^{x+1}^{}.

## I almost forgot: If *x* is negative, then you'll get a number that is greater than 1½, which will most likely be an integer instead of a non-integer!

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© Derek Cumberbatch