Arithmetic
Essential formulas and concepts in arithmetic
Sum of the first $n$ natural numbers
\(1 + 2 + \cdots + n = \frac{n(n+1)}{2}\)
Variables:
- \(n is a positive integer.\)
Description/Usage:
Gives the sum of all natural numbers from \(1\) up to \(n\). Often illustrated by pairing terms (e.g. \(1+n\), \(2+(n-1)\), etc.) to simplify the addition.
Mathematical concept visualization
Arithmetic sequence (nth term)
\(a_n = a_1 + (n-1)d\)
Variables:
- \(a_n is the nth term of the sequence, a_1 is the first term, d is the common difference, and n is the term index.\)
Description/Usage:
Defines the general term of an arithmetic progression (constant difference between consecutive terms). A linear graph represents the sequence values.
Mathematical concept visualization
Arithmetic series (sum of first $n$ terms of an AP)
\(S_n = \frac{n}{2}(2a_1 + (n-1)d)\)
Variables:
- \(S_n is the sum of the first n terms, a_1 is the first term, d is the common difference, n is the number of terms.\)
Description/Usage:
Calculates the sum of an arithmetic progression. Often visualized by pairing terms from ends toward the center (which all yield the same sum).
Mathematical concept visualization