QuestionFind the sum. $\quad \sum_{n=3}^{7} 2^{n}$

Studdy Solution

STEP 1

What is this asking? We need to find the sum of the powers of 2, starting from $2^3$ and going all the way up to $2^7$ . Watch out! Don't start at $2^1$ or $2^2$ !
We're starting at $2^3$ .
Also, don't forget to add the powers of 2, not multiply them.

STEP 2

1. Expand the Sum
2. Calculate the Powers
3. Add the Powers

STEP 3

Let's expand this fancy summation notation into something a little more down-to-earth.
The sigma, $\sum$ , just means "sum up a bunch of stuff." The $n=3$ at the bottom tells us our starting value for $n$ is 3, and the $7$ at the top tells us where to stop!

STEP 4

So, $\sum_{n=3}^{7} 2^{n}$ means we're summing up $2^n$ for each value of $n$ from 3 to 7.
Let's write it out: $2^3 + 2^4 + 2^5 + 2^6 + 2^7$ .
Much better!

STEP 5

Now, let's calculate each of these powers of 2: $2^3 = 2 \cdot 2 \cdot 2 = 8$ $2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16$ $2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$ $2^6 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 64$ $2^7 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 128$

STEP 6

Almost there!
Now we just need to add up all the values we just calculated: $8 + 16 + 32 + 64 + 128$

STEP 7

Let's do this carefully: $8 + 16 = 24$ $24 + 32 = 56$ $56 + 64 = 120$ $120 + 128 = 248$

STEP 8

So, the sum $\sum_{n=3}^{7} 2^{n}$ is equal to 248!

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