The value of 11² + 12² + 13² + . . . . . . + 20² is:

Formula Used:

Sum of square of n numbers = \(\frac{n(n + 1)(2n + 1)}{6}\) 

Calculation:   

Case: 1

12 + 22 + 32 + . . . . . . + 202

Sum of square of 20 numbers = \(\frac{20(20 + 1)(2\times 20\space + \space 1)}{6}\)

⇒ Sum of square of 20 numbers = \(\frac{20 \times 21\times 41}{6}\)

⇒ Sum of square of 20 numbers = 2870

Case: 2

 12 + 22 + 32 + . . . . . . + 102

⇒ Sum of square of 10 numbers = \(\frac{10(10 + 1)(2\times 10\space + \space 1)}{6}\)

⇒ Sum of square of 10 numbers = \(\frac{10 \times 11\times 21}{6}\)

⇒ Sum of square of 10 numbers = 385

According to the question: 

⇒ 112 + 122 + 132 + . . . . . . + 202 = Sum of square of 20 numbers - Sum of square of 10 numbers 

⇒ 112 + 122 + 132 + . . . . . . + 202 = 2870 - 385 

∴ 112 + 122 + 132 + . . . . . . + 202 = 2485