Simplify:
(x-1/2)-(x-2/2)+(x-3/2)-...-(x-100/2)=
(x-1/2)-(x-2/2)+(x-3/2)-...-(x-100/2)
= x - 1/2 - x + 2/2 + x - 3/2 - .... - x + 100/2
= x-x+x-x+x-....-x -1/2 + 2/2 - 3/2 - ... + 100/2
looking at the last term, we see there are 100 terms, so the x's drop out
= -1/2 + 2/2 - 3/2 - ... + 100/2
= (-1/2)(1 - 2 + 3 - .... + 99 - 100)
= (-1/2)[ (1-2) + (3-4) + ... + (99-100]
= (-1/2)[ 50(-1)]
= 25
Not claiming that this is the shortest or best way of doing it, but it is easy to understand
To simplify the given expression, we need to combine like terms. Let's break down the expression step by step:
First, let's rewrite the terms in a more simplified form:
(x - 1/2) = (2x/2 - 1/2) = (2x - 1)/2
(x - 2/2) = (2x/2 - 2/2) = (2x - 2)/2
(x - 3/2) = (2x/2 - 3/2) = (2x - 3)/2
We can continue this pattern until we reach the last term:
(x - 100/2) = (2x/2 - 100/2) = (2x - 100)/2
Now, we can rewrite the given expression using the simplified forms:
(2x - 1)/2 + (2x - 2)/2 + (2x - 3)/2 + ... + (2x - 100)/2
Next, we need to distribute the division by 2 to each term:
(2x/2 - 1/2) + (2x/2 - 2/2) + (2x/2 - 3/2) + ... + (2x/2 - 100/2)
Now, we can combine like terms:
(2x + 2x + 2x + ... + 2x)/2 - (1/2 + 2/2 + 3/2 + ... + 100/2)
Since we have 100 terms with 2x in each, we can simplify the numerator:
(200x)/2 - (1/2 + 2/2 + 3/2 + ... + 100/2)
Further simplifying,
100x - (1/2 + 2/2 + 3/2 + ... + 100/2)
To find the sum of the terms from 1/2 to 100/2, we can use the formula for the sum of an arithmetic series:
Sum = (n/2) * (first term + last term)
where n is the number of terms.
In this case, we have a series with a common difference of 1/2, a first term of 1/2, and a last term of 100/2 (which is 50).
n = 50 - 1/2 + 1 = 50.
Substituting this into the formula:
sum = (50/2) * (1/2 + 50/2) = (25) * (1/2 + 50/2) = (25) * (51/2) = (25)(51) = 1275
Therefore, our simplified expression is:
100x - 1275
(x-1/2)-(x-2/2)+(x-3/2)-...-(x-100/2)
x-x+x-x+...+x-x - 1/2 + 2/2 - 3/2 + ... - 99/2 + 100/2
the x-x+x-x terms cancel out, and we are left with
1/2 (-1+2 -3+4 ... -99+100)
= 1/2 (1+1...+1) (50 times)
= 1/2 * 50
= 25