Beauty+of+Math

__ **Beauty of mathematics** __


5 = 3 + 2 = 3² – 2² 7 = 4 + 3 = 4² – 3² 9 = 5 + 4 = 5² – 4² 11 = 6 + 5 = 6² – 5² Etc…
 * __Odd numbers__**

1² = 1³ = 1 (1 + 2)² = 1³ + 2³ = 9 (1 + 2 + 3)² = 1³ + 2³ + 3³ = 36 (1 + 2 + 3 + 4)² = 1³ + 2³ + 3³ + 4³ = 100 (1 + 2 + 3 + 4 + 5)² = 1³ + 2³ + 3³ + 4³ + 5³ = 225 (1 + 2 + 3 + 4 + 5 + 6)² = 1³ + 2³ + 3³ + 4³ + 5³ + 6³ = 441 ...
 * __Square of a sum & sum of cubes__**


 * 1 ** **6** = 4 ²
 * 1 **1 5 **6** = 3 4 ²
 * 1 **1 1 5 5 **6** = 33 4 ²
 * 1 **1 1 1 5 5 5 **6** = 333 4 ²
 * 1 **1 1 1 1 5 5 5 5 **6** = 3333 4 ²

Each number in this sequence is obtained from the previous one by inserting 15 in the middle like the extra leaf in an expanding table. Can you show why the numbers in this sequence will always be square no matter how far you go?

There is one other sequence of numbers like this one, see below :
 * 4 ** **9** = 7 ²
 * 4 **4 8 **9** = 6 7 ²
 * 4 **4 4 8 8 **9** = 66 7 ²
 * 4 **4 4 4 8 8 8 **9** = 666 7 ²
 * 4 **4 4 4 4 8 8 8 8 **9** = 6666 7 ²

1 × **8** + 1 = 9 12 × **8** + 2 = 98 123 × **8** + 3 = 987 1234 × **8** + 4 = 9876 12345 × **8** + 5 = 98765 123456 × **8** + 6 = 987654 1234567 × **8** + 7 = 9876543 12345678 × **8** + 8 = 98765432 123456789 × **8** + 9 = 987654321

1 × **9** + 2 = 11 12 × **9** + 3 = 111 123 × **9** + 4 = 1111 1234 × **9** + 5 = 11111 12345 × **9** + 6 = 111111 123456 × **9** + 7 = 1111111 1234567 × **9** + 8 = 11111111 12345678 × **9** + 9 = 111111111 123456789 × **9** + 10 = 1111111111

9 × **9** + 7 = 88 98 × **9** + 6 = 888 987 × **9** + 5 = 8888 9876 × **9** + 4 = 88888 98765 × **9** + 3 = 888888 987654 × **9** + 2 = 8888888 9876543 × **9** + 1 = 88888888 98765432 × **9** + 0 = 888888888

Brilliant, isn't it? Take a look at this symmetry: 1 × 1 = **1** 11 × 11 = 1 **2** 1 111 × 111 = 12 **3** 21 1111 × 1111 = 123 **4** 321 11111 × 11111 = 1234 **5** 4321 111111 × 111111 = 12345 **6** 54321 1111111 × 1111111 = 123456 **7** 654321 11111111 × 11111111 = 1234567 **8** 7654321 111111111 × 111111111 = 12345678 **9** 87654321