QuestionQuestion 5
Watch the video that describes using summation notation.
Click here to watch the video.
Evaluate the sum.
(Simplify your answer.)
Studdy Solution
STEP 1
What is this asking? We need to calculate the sum of for each integer from 1 to 10. Watch out! Don't forget to add 3 for *each* value of .
STEP 2
1. Expand the Summation
2. Regroup the Terms
3. Calculate the Sum
STEP 3
Alright, let's **expand** this summation!
We're going from to , so we'll have ten terms in total.
Each term is , so let's write them all out:
STEP 4
See how we substituted each value of into the expression ?
This gives us a clear picture of what we're adding up.
STEP 5
Now, let's **regroup** the terms to make the calculation easier.
We'll group all the terms together and all the 3's together.
STEP 6
Notice that we have ten 3's added together.
This is because we added 3 for each of the ten values of .
STEP 7
Let's **calculate** the sum of the terms first.
This is the sum of the first ten natural numbers, which we can calculate using the formula:
where is the **last number** in the sequence.
In our case, , so the sum is:
STEP 8
Now, let's **calculate** the sum of the 3's.
We have ten 3's, so their sum is:
STEP 9
Finally, let's **add** these two sums together to get our **final result**:
STEP 10
The sum is .
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