The sum of all the integers from 100 to 1000, inclusive, is x. What expression represents the sum of all integers from 104 to 1000, inclusive?
The correct answer and explanation is:
Let’s analyze the problem carefully.
Problem Recap:
- The sum of all integers from 100 to 1000, inclusive, is xx.
- We want to find an expression that represents the sum of all integers from 104 to 1000, inclusive.
Step 1: Understand the sum xx
x=∑k=1001000kx = \sum_{k=100}^{1000} k
This means xx is the total sum starting at 100 and going up to 1000.
Step 2: Express the sum from 104 to 1000 in terms of xx
The sum from 104 to 1000 is: ∑k=1041000k\sum_{k=104}^{1000} k
Since xx includes all integers from 100 to 1000, the difference between xx and the sum from 104 to 1000 is the sum of the integers from 100 to 103: ∑k=100103k=100+101+102+103\sum_{k=100}^{103} k = 100 + 101 + 102 + 103
So, ∑k=1041000k=x−∑k=100103k\sum_{k=104}^{1000} k = x – \sum_{k=100}^{103} k
Step 3: Calculate the sum of integers from 100 to 103
100+101+102+103=(100+103)+(101+102)=203+203=406100 + 101 + 102 + 103 = (100 + 103) + (101 + 102) = 203 + 203 = 406
Final Expression:
∑k=1041000k=x−406\boxed{ \sum_{k=104}^{1000} k = x – 406 }
Explanation (300 words):
When you have the sum of all integers from 100 to 1000 (inclusive), denoted by xx, it includes every number in that range added together. If you want to find the sum of integers starting from 104 up to 1000, you can use the sum xx and subtract the part that you no longer want to include, which is from 100 up to 103.
Think of xx as a big sum: x=100+101+102+103+104+⋯+1000x = 100 + 101 + 102 + 103 + 104 + \cdots + 1000
If we remove the first four numbers (100, 101, 102, 103) from xx, the remaining sum will be from 104 to 1000: x−(100+101+102+103)=∑k=1041000kx – (100 + 101 + 102 + 103) = \sum_{k=104}^{1000} k
Calculating the sum of 100 to 103, we get 406. Thus, the expression becomes x−406x – 406.
This approach uses the property of sums over consecutive integers — you can always break down or subtract parts of sums if you know the total sum for a larger range. This is helpful because it lets you use xx, which is already given, without needing to recompute the sum from scratch.
If you wanted to find xx directly, you would use the formula for the sum of an arithmetic series: Sum=n2×(first term+last term)\text{Sum} = \frac{n}{2} \times (\text{first term} + \text{last term})
where nn is the number of terms. But since the problem defines xx as the sum from 100 to 1000, you don’t need to calculate it again — just subtract the known small part.
Therefore, the sum from 104 to 1000 is x−406x – 406.