Square Pyramidal Number Calculator

Understanding Square Pyramidal Numbers

What are Square Pyramidal Numbers?

Square pyramidal numbers are a sequence of numbers that represent the number of objects needed to form a square-based pyramid. Each layer of the pyramid is a square, and the sequence is formed by summing the squares of the first n positive integers.

Mathematical Definition

The nth square pyramidal number, denoted as \(P_n\), is given by the formula:

\[P_n = \sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}\]

Where:

• \(P_n\) is the nth square pyramidal number
• \(n\) is the number of layers in the pyramid

Calculation Method

To calculate the nth square pyramidal number:

1. Substitute the value of n into the formula
2. Multiply n, (n+1), and (2n+1)
3. Divide the result by 6

Example Calculation

Let's calculate the 4th square pyramidal number:

1. \(P_4 = \frac{4(4+1)(2\cdot4+1)}{6}\)
2. \(P_4 = \frac{4 \cdot 5 \cdot 9}{6}\)
3. \(P_4 = \frac{180}{6} = 30\)

Visual Representation

This diagram represents the 4th square pyramidal number as a pyramid with 4 layers, totaling 30 units.