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Novosadov [1.4K]
3 years ago
13

What is the value of 4*(-1+2-3+4-3+6-7+....+100) Show your work. [Pls help I am really drowning in work. 50 points + brainliest

for first answer]
Mathematics
1 answer:
KengaRu [80]3 years ago
8 0

Answer:

200

Step-by-step explanation:

We have:

4(-1+2-3+4-5+6-7...+100)

We can rearrange the numbers to obtain:

4((-1-3-5-7-...-99)+(2+4+6...+100))

From the left, we can factor out a negative. So:

4(-(1+3+5+7+...+99)+(2+4+6...+100))

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:

(1+3+5+7+...+99)

We can use the arithmetic series formula, where:

S=\frac{k}{2}(a+x_k)

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:

S=\frac{50}{2}(1+99})

Divide the fraction. Add within the parentheses:

S=25(100)

Multiply:

S=2500

So, the sum of all the odd terms is 2500.

Even Terms:

We have:

(2+4+6+...+100)

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:

S=\frac{50}{2}(2+100)

Divide and add:

S=25(102)

Multiply:

S=2550

We originally had:

4(-(1+3+5+7+...+99)+(2+4+6...+100))

Substitute them for their respective sums:

4(-(2500)+2550)

Multiply:

4(-2500+2550)

Add:

=4(50)

Multiply:

=200

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:

4\cdot(-1+2-3+4-5+6-7+...+100)

Let's group every two terms together. So:

=4((-1+2)+(-3+4)+(-5+6)...+(-99+100))

We can see that they each sum to 1:

=4((1)+(1)+(1)+...+(1))

Since there are 100 terms, we will have 50 pairs, so 50 times 1. So:

=4(50)

Multiply:

=200

Pick which one you want to use! I will suggest this one though...

Edit: Typo

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