MBMA-027 ### Step 1: Understanding the Problem
Before diving into solving the problem, it’s important to thoroughly understand what it entails. The problem requires calculating the sum of all integers from 1 to 100, then calculating the sum of all integers from either 1 to 50 or 51 to 100, and finally finding the difference between these two sums.
At this stage, I’m still unsure about which range to choose for the second sum: from 1 to 50 or from 51 to 100. My initial thought is to choose from 1 to 50 for the second sum because it seems simpler to start lower, but I’m not entirely sure if this will yield the desired result.
### Step 2: Calculating the Sum from 1 to 100
The first step is to calculate the sum of all integers from 1 to 100. One effective method for this is using 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, 100)
- **first term** is the first term in the series ( 1 )
- **last term** is the last term in the series ( 99 )
Using this formula:
**Sum= (100/2) × (1 + 100) = 50 × 101 = 5050**
So, the sum from 1 to 100 is 5050.
### Step 3: Calculating the Sum from 1 to 50
Next, I need to calculate the sum from 1 to 50. I’ll again 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 50 )
- **first term** is the first person in the series ( 1 )
- **last term** is the last term in the series ( 50 )
Using this formula:
**Sum= (50/2) × (1 + 50) = 25 × 51 = 1275**
So, the sum from 1 to 50 is 1275.
### Step 4: Calculating the Difference
Now, using the two sums calculated, I’ll find the difference between them.
**Difference = Sum from 1 to 100 - Sum from 1 to 50 = 5050 - 1275 = 3775**
So, the final answer is 3775.
### Verification: Final Answer
Let’s verify the answer by summing the numbers from 51 to 100:
**Sum= (n/2) × (first term + last term)**
Where:
- **n** is the number of terms ( in this case 50 )
- **first term** is the first term in the series ( 51 )
- **last term** is the last term in the series ( 100 )
Using this formula:
**Sum= (90/2) × (51 + 100) = 45 × 151 = 6795
Wait, this doesn’t seem right. Let me recalculate.
It should be:
**Sum= (50/2) × (51 + 100) = 25 × 151 = 3775**
So, the sum from 51 to 100 is 3775, which matches our previous difference of 3775. Thus, the final answer is 3775.
**Conclusion: What is the difference between the sum of all integers from 1 to 100 and the sum of all integers from 51 to 100?**
The difference stands at 3775.
2025年4月18日
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