1. What is the Sum of Consecutive Integers?

The sum of consecutive integers is the total obtained by adding a sequence of integers where each term increases by 1 from the previous term. This is a fundamental concept in arithmetic and number theory.

2. How Does the Calculator Work?

The calculator uses the arithmetic series sum formula:

Where for consecutive integers:

• \( a \) — First term in the sequence
• \( n \) — Number of terms to sum
• \( d \) — Common difference (always 1 for consecutive integers)

Explanation: The formula efficiently calculates the sum without needing to add each term individually, saving time especially for large sequences.

3. Importance of Sum Calculation

Details: Calculating sums of consecutive integers is important in various mathematical applications including series analysis, algorithm design, and solving real-world problems involving sequential numbering or counting.

4. Using the Calculator

Tips: Enter the first integer in your sequence and the number of consecutive integers you want to sum. Both values must be integers (n must be positive).

5. Frequently Asked Questions (FAQ)

Q1: Can this calculator handle negative integers?
A: Yes, the calculator works for any integer starting value, positive or negative.

Q2: What's the largest sequence this can calculate?
A: The calculator can handle very large sequences, limited only by PHP's integer size (typically up to 2^63-1 on 64-bit systems).

Q3: How is this different from triangular numbers?
A: Triangular numbers are a special case where the sequence starts at 1 (1+2+3+...+n).

Q4: Can I calculate partial sums with this?
A: Yes, you can sum any consecutive portion of integers by specifying the starting value and count.

Q5: Is there a formula for non-consecutive sequences?
A: Yes, the general arithmetic series formula works for any common difference d between terms.