The sequence \{x_n\}_{n=1}^{n=20} is defined as x_n = (-1)^n n. What is the sum of all the terms in the sequence?
To find the sum of all the terms in the sequence, we can simply add up all the terms. Let's calculate each term and then sum them up.
The sequence is defined as x_n = (-1)^n * n for n = 1 to 20.
We can see that the sequence alternates between positive and negative values based on the value of n. When n is even, (-1)^n is equal to 1, and when n is odd, (-1)^n is equal to -1.
Now, let's calculate each term:
For n = 1, x_1 = (-1)^1 * 1 = -1 * 1 = -1
For n = 2, x_2 = (-1)^2 * 2 = 1 * 2 = 2
For n = 3, x_3 = (-1)^3 * 3 = -1 * 3 = -3
For n = 4, x_4 = (-1)^4 * 4 = 1 * 4 = 4
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and so on, up to n = 20.
Now, let's calculate the sum of all the terms:
Sum = x_1 + x_2 + x_3 + ... + x_19 + x_20
Since the sequence alternates between positive and negative values, we can simplify the sum:
Sum = (-1) + 2 + (-3) + 4 + ... + (-19) + 20
To simplify this sum, we can group the terms in pairs:
Sum = (-1 + 2) + (-3 + 4) + ... + (-19 + 20)
Each pair of terms results in a positive number, so the sum further simplifies to:
Sum = 1 + 1 + 1 + ... + 1
There are 20 terms in the sequence, and each term is equal to 1. Therefore, the sum can be calculated as:
Sum = 20 * 1 = 20
So, the sum of all the terms in the sequence is 20.