Math Problem Statement
Solution
The expression shown is:
This is a formula for the sum of the first even numbers. We can verify the formula by understanding that each term in the series is an even number, , and we need to find the sum of these terms up to .
Explanation:
- General Form of Sum of Even Numbers: The -th even number can be written as .
- Sum of Series: The series can be represented as:
- Sum of First Natural Numbers: The sum of the first natural numbers is:
- Final Calculation: Substituting this result back, we get: which confirms the formula.
Thus, the sum of the first even numbers is indeed .
Would you like further details or have any questions?
Further Questions
- What is the formula for the sum of the first odd numbers?
- How can you generalize this approach to find the sum of multiples of a different integer, such as 3?
- Can we use a similar approach to find the sum of squares of the first even numbers?
- How does this formula relate to arithmetic series in general?
- What other techniques can be used to verify this summation formula?
Tip
Remember that finding the sum of an arithmetic series often involves identifying a pattern and using known formulas for series, such as the sum of the first natural numbers.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Algebra
Series and Sequences
Arithmetic Series
Formulas
Sum of first n even numbers: 2 + 4 + 6 + ... + 2n = n(n + 1)
Sum of first n natural numbers: 1 + 2 + 3 + ... + n = n(n + 1) / 2
Theorems
Arithmetic series formula
Suitable Grade Level
Grades 9-10
Related Recommendation
Mathematical Induction Proof of the Sum of First n Even Numbers
Sum of the First 24 Even Numbers
Calculate the Sum of the First 100 Positive Odd Numbers
Mathematical Induction: Proving Summation of Even Numbers 2 + 4 + 6 + ... + 2n = n(n+1)
Sum of All Even Numbers Between 1 and 51 Using Arithmetic Sequence Formula