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# Square 1 to 50 — List of Perfect Square Values

[#Algebra](/content/tag/algebra/index.html)

TL;DR

The squares from 1 to 50 range from 1^2 = 1 to 50^2 = 2500. This article gives the complete list, the patterns that make memorisation faster (the last-digit pattern, the squares-near-50 trick), and the most common contexts where these values appear in school maths.

## The Answer in One Line

The squares from 1 to 50 are 1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,400,441,484,529,576,625,676,729,784,841,900,961,1024,1089,1156,1225,1296,1369,1444,1521,1600,1681,1764,1849,1936,2025,2116,2209,2304,2401,2500.

The first ten (1^2 to 10^2) appear in nearly every algebra problem.

## Quick Answer

> **Result:** the squares from 1^2 = 1 to 50^2 = 2500.
> **Notation:** n^2 = n \times n.
> **Method shown:** direct multiplication and the squares-near-50 shortcut.
> **Pattern:** the last digit of n^2 depends only on the last digit of n (specifically the pattern 0,1,4,9,6,5,6,9,4,0).

## Quick Reference Table — Squares 1 to 50

| n | n^2 | n | n^2 | n | n^2 | n | n^2 | n | n^2 |
|---|-----|---|-----|---|-----|---|-----|---|-----|
| 1 | 1   | 11| 121 | 21| 441 | 31| 961 | 41| 1681 |
| 2 | 4   | 12| 144 | 22| 484 | 32| 1024| 42| 1764 |
| 3 | 9   | 13| 169 | 23| 529 | 33| 1089| 43| 1849 |
| 4 | 16  | 14| 196 | 24| 576 | 34| 1156| 44| 1936 |
| 5 | 25  | 15| 225 | 25| 625 | 35| 1225| 45| 2025 |
| 6 | 36  | 16| 256 | 26| 676 | 36| 1296| 46| 2116 |
| 7 | 49  | 17| 289 | 27| 729 | 37| 1369| 47| 2209 |
| 8 | 64  | 18| 324 | 28| 784 | 38| 1444| 48| 2304 |
| 9 | 81  | 19| 361 | 29| 841 | 39| 1521| 49| 2401 |
| 10| 100 | 20| 400 | 30| 900 | 40| 1600| 50| 2500 |

## Where the Squares 1 to 50 Appear

The squares from 1 to 50 are the values most students see most often. They appear in the Pythagorean theorem (a leg of length 7 and a leg of length 24 produce a hypotenuse \( \sqrt{49 + 576} = \sqrt{625} = 25\)), in quadratic-equation solutions (every perfect-square discriminant in the table makes a quadratic factor over the integers), and in any SAT or ACT mental-math section where speed depends on knowing the table by heart.

## Concept Definition — Perfect Squares

A **perfect square** is an integer that equals some integer multiplied by itself. The square of n is n^2 = n \cdot n.

For positive integers, n^2 is also the area of a square with side length n. The geometric interpretation is built into the name.

## How to Compute Squares Quickly

### **Method 1 — Direct multiplication**

n \times n.

For two-digit n: \(34^2 = 34 \times 34 = 1156\) (use long multiplication).

### **Method 2 — The squares-near-50 trick**

For n=50±k:
n^2=2500+100k\cdot(sign)+k^2.

### **Example.**  \(47^2=(50−3)^2=2500−300+9=2209\).

### **Method 3 — Squares of multiples of 5**

\((5n)^2=25n^2\).

### **Method 4 — Last-digit pattern**

The last digit of n^2 follows the cycle 0,1,4,9,6,5,6,9,4,0.

## Common Mistakes With Square 1 to 50

### **1. Confusing n^2 with 2n**

**Don't do this:** n^2 = 2n.

**The correct way:** n^2 = n \times n.

### **2. Off-by-one in the table.**

**Don't do this:** Trust memory without checking neighbours.

**The correct way:** 132 = 169. Adjacent squares differ by 2n + 1 — useful sanity check.

## Frequently Asked Questions

What are the perfect squares between 1 and 50?

The integers between 1 and 50 that are perfect squares: 1,4,9,16,25,36,49 — that's 1^2 through 7^2.

What is the sum of squares from 1 to 50?

\(\sum_{k=1}^{50} k^2 = \frac{50 \cdot 51 \cdot 101}{6} = 42{,}925\).

What is the largest perfect square below 2500?

49^2 = 2401. The next one (50^2 = 2500) reaches exactly the boundary.

Is the square of an odd number always odd?

Yes. Odd × odd = odd.

What is 50^2?

50^2 = 2500.