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Overview

Relations Among Sequences

April 10, 2024
1 min read
odd-numbers square-numbers summation

Surprising Relationships

Number sequences are often related to each other. One of the most famous examples is the relationship between Odd Numbers and Square Numbers.

Adding Odd Numbers

What happens when we add consecutive odd numbers starting from 1?

1=1=121+3=4=221+3+5=9=321+3+5+7=16=421+3+5+7+9=25=52\begin{aligned} 1 &= 1 &= 1^2 \\ 1 + 3 &= 4 &= 2^2 \\ 1 + 3 + 5 &= 9 &= 3^2 \\ 1 + 3 + 5 + 7 &= 16 &= 4^2 \\ 1 + 3 + 5 + 7 + 9 &= 25 &= 5^2 \end{aligned}

Conclusion: The sum of the first nn odd numbers is equal to the nn-th square number (n2n^2).

Visual Proof

We can visualize this by taking a square grid and “peeling” off L-shaped layers. Each layer contains an odd number of dots.

Adding Up and Down

Another way to generate square numbers is by adding counting numbers up to a peak and then back down to 1.

1=1=121+2+1=4=221+2+3+2+1=9=321+2+3+4+3+2+1=16=42\begin{aligned} 1 &= 1 &= 1^2 \\ 1 + 2 + 1 &= 4 &= 2^2 \\ 1 + 2 + 3 + 2 + 1 &= 9 &= 3^2 \\ 1 + 2 + 3 + 4 + 3 + 2 + 1 &= 16 &= 4^2 \end{aligned}

This visualizes the square tilted on its corner, or “stacked” diagonally.