Find the measure of
1 answer:
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Answer:
200
Step-by-step explanation:
We have:
We can rearrange the numbers to obtain:
From the left, we can factor out a negative. So:
In other words, we want to find the sum of all the odd numbers from 1 to 99.
And the sum of all the even numbers from 2 to 100.
Let's do each one individually:
Odd Terms:
We have:
We can use the arithmetic series formula, where:
Where k is the number of terms, a is the first term, and x_k is the last term.
Since it's all the odd numbers between 1 and 99, there are 50 terms.
Our first term is 1 and our last term is 99. So, the sum of all the odd terms are:
Divide the fraction. Add within the parentheses:
Multiply:
So, the sum of all the odd terms is 2500.
Even Terms:
We have:
Again, we can use the above formula.
Our first term is 2, last term is 100. And since it's from 2-100, we have 50 even terms. So:
Divide and add:
Multiply:
We originally had:
Substitute them for their respective sums:
Multiply:
Add:
Multiply:
So, the sum of our sequence is 200.
And we're done!
Note: I just found a <em>way</em> easier way to do this. We have:
Let's group every two terms together. So:
We can see that they each sum to 1:
Since there are 100 terms, we will have 50 pairs, so 50 times 1. So:
Multiply:
Pick which one you want to use! I will suggest this one though...
Edit: Typo
Answer: 26.8 feet
Step-by-step explanation:
In the figure attached you can see two right triangles triangle ABD and a triangle ACD.
You are located at point B and the other person at point C.
The approximate height of the lifeguard station is <em>x</em>.
Keep on mind that:
Therefore:
<em>For the triangle ABD</em>:
[EQUATION 1]
<em>For the triangle ACD:</em>
[EQUATION 2]
Solve from DC from [EQUATION 2]:
Substitute into [EQUATION 1] and solve for x:
≈26.8ft
