Question $2^{10}+2^{11}+2^{12}+\ldots+2^{2004} \rightarrow 2^{n}-2^{m}$ $m, n=$ natural numbers

Studdy Solution

STEP 1

What is this asking? We're asked to rewrite the sum of powers of 2, from $2^{10}$ all the way up to $2^{2004}$ , in the much simpler form of $2^n - 2^m$ , where n and m are natural numbers. Watch out! It's easy to mess up the powers when factoring.
Double-check each step carefully!

STEP 2

1. Factor the expression
2. Simplify using the geometric series formula
3. Express in the desired form

STEP 3

Alright, let's factor out the smallest power of 2 from our sum, which is $2^{10}$ !
This is like reverse distributing, and it's a super useful trick!

STEP 4

$2^{10} + 2^{11} + 2^{12} + \ldots + 2^{2004} = 2^{10}(1 + 2^1 + 2^2 + \ldots + 2^{1994})$ See how neat that is?
We've got $2^{10}$ multiplied by a much simpler sum inside the parentheses.

STEP 5

Now, peep this: the sum inside the parentheses $1 + 2^1 + 2^2 + \ldots + 2^{1994}$ is a geometric series!
And we've got a handy-dandy formula for that.
Remember, a geometric series is a sum where each term is multiplied by the same value (the common ratio) to get the next term.
Here, our common ratio is 2.

STEP 6

The formula for the sum of a geometric series is $\frac{a(r^n - 1)}{r - 1}$ , where a is the first term (here, $a = 1$ ), r is the common ratio (here, $r = 2$ ), and n is the number of terms.
How many terms do we have?
Well, the powers of 2 go from 0 to 1994, so we have $1994 - 0 + 1 = \textbf{1995}$ terms.

STEP 7

Plugging these values into our formula, we get: $1 + 2^1 + 2^2 + \ldots + 2^{1994} = \frac{1 \cdot (2^{1995} - 1)}{2 - 1} = \frac{2^{1995} - 1}{1} = 2^{1995} - 1$ So, our original expression becomes: $2^{10}(2^{1995} - 1)$

STEP 8

Let's distribute that $2^{10}$ back in: $2^{10}(2^{1995} - 1) = 2^{10} \cdot 2^{1995} - 2^{10} \cdot 1 = 2^{10 + 1995} - 2^{10} = 2^{2005} - 2^{10}$

STEP 9

Boom! We've got it in the form $2^n - 2^m$ , where $n = \textbf{2005}$ and $m = \textbf{10}$ .

STEP 10

$m = 10$ and $n = 2005$

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms