Math Problem Statement

what a solution 1+2+3+... +198+199+200=? show your solution

Solution

To solve the sum 1+2+3++198+199+2001 + 2 + 3 + \dots + 198 + 199 + 200, we can use the formula for the sum of the first nn natural numbers:

Sn=n(n+1)2S_n = \frac{n(n+1)}{2}

In this problem, n=200n = 200, so we can substitute n=200n = 200 into the formula:

S200=200×(200+1)2S_{200} = \frac{200 \times (200 + 1)}{2}

Simplifying this:

S200=200×2012=402002=20100S_{200} = \frac{200 \times 201}{2} = \frac{40200}{2} = 20100

Thus, the sum 1+2+3++198+199+2001 + 2 + 3 + \dots + 198 + 199 + 200 is 20100\boxed{20100}.

Would you like more details or have any questions?

Here are 8 related questions for further exploration:

  1. What is the sum of the first 100 natural numbers?
  2. How would you find the sum of all even numbers from 2 to 200?
  3. Can you prove the formula Sn=n(n+1)2S_n = \frac{n(n+1)}{2} using induction?
  4. What is the sum of the first 200 odd numbers?
  5. How would you calculate the sum of squares of the first 200 natural numbers?
  6. How does the sum formula change if the sequence starts from a number other than 1?
  7. What is the sum of the first nn multiples of a given number?
  8. How would you approach finding the sum 1+2+3++9991 + 2 + 3 + \dots + 999?

Tip: For large sums or sequences, look for patterns or use known formulas to simplify calculations.

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Math Problem Analysis

Mathematical Concepts

Arithmetic
Summation

Formulas

Sum of first n natural numbers: S_n = n(n+1)/2

Theorems

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Suitable Grade Level

Grades 7-9