Given that 1+2+3+...100= 100(1+100)/2.
Simplify (x+1)+2(x+1)+...+1000(x+1)
well, that's
(x+1)(1+2+3+...+1000), right?
To simplify the given expression: (x+1) + 2(x+1) + ... + 1000(x+1), we can first simplify each term individually, and then combine them.
Each term in the expression can be written as (x+1) multiplied by a number from 1 to 1000.
Let's start by factoring out (x+1) from each term:
(x+1) + 2(x+1) + ... + 1000(x+1) = (x+1) (1 + 2 + ... + 1000)
Now, let's use the formula you provided to find the sum of the numbers from 1 to 1000:
1 + 2 + 3 + ... + 1000 = 1000(1 + 1000)/2 = 1000(1001)/2 = 500 * 1001 = 500500
Substituting this value back into the expression:
(x+1) + 2(x+1) + ... + 1000(x+1) = (x+1) * 500500
Therefore, the simplified expression is 500500(x+1).