Understanding the Sum of Squares Problem

The question asks for the value of the sum of squares of integers from 10 to 19, which can be written as $10^2 + 11^2 + 12^2 + \dots + 19^2$. This is a specific example of calculating the sum of squares for a sequence of consecutive numbers that don't start from 1.

To solve this, we can use the known formula for the sum of squares of the first \(n\) natural numbers.

Formula for the Sum of Squares

The formula for the sum of squares of the first \(n\) positive integers is given by:

$\sum_{k=1}^{n} k^2 = 1^2 + 2^2 + 3^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}$

We need to find the sum from 10 to 19. We can achieve this by finding the sum of squares from 1 to 19 and subtracting the sum of squares from 1 to 9. This is because:

$(10^2 + 11^2 + \dots + 19^2) = (1^2 + 2^2 + \dots + 19^2) - (1^2 + 2^2 + \dots + 9^2)$

Calculating the Sum of Squares up to 19

Using the formula with \(n=19\):

$\sum_{k=1}^{19} k^2 = \frac{19(19+1)(2 \times 19+1)}{6}$

$\sum_{k=1}^{19} k^2 = \frac{19(20)(38+1)}{6}$

$\sum_{k=1}^{19} k^2 = \frac{19 \times 20 \times 39}{6}$

Now, we simplify the expression:

$\sum_{k=1}^{19} k^2 = \frac{19 \times (20/2) \times (39/3)}{(6/6)}$

$\sum_{k=1}^{19} k^2 = 19 \times 10 \times 13$

$\sum_{k=1}^{19} k^2 = 190 \times 13$

$\sum_{k=1}^{19} k^2 = 2470$

Calculating the Sum of Squares up to 9

Using the formula with \(n=9\):

$\sum_{k=1}^{9} k^2 = \frac{9(9+1)(2 \times 9+1)}{6}$

$\sum_{k=1}^{9} k^2 = \frac{9(10)(18+1)}{6}$

$\sum_{k=1}^{9} k^2 = \frac{9 \times 10 \times 19}{6}$

Now, we simplify the expression:

$\sum_{k=1}^{9} k^2 = \frac{(9/3) \times (10/2) \times 19}{(6/6)}$

$\sum_{k=1}^{9} k^2 = 3 \times 5 \times 19$

$\sum_{k=1}^{9} k^2 = 15 \times 19$

$\sum_{k=1}^{9} k^2 = 285$

Finding the Desired Sum

To find the sum of squares from 10 to 19, we subtract the sum from 1 to 9 from the sum from 1 to 19:

Sum ($10^2$ to $19^2$) = $\sum_{k=1}^{19} k^2 - \sum_{k=1}^{9} k^2$

Sum ($10^2$ to $19^2$) = $2470 - 285$

Sum ($10^2$ to $19^2$) = $2185$

Summary of Calculation

Thus, the value of $10^2 + 11^2 + 12^2 + \dots + 19^2$ is 2185. This method effectively calculates the sum of squares for a non-starting sequence.

Understanding the formula for the sum of squares is key to solving this type of problem efficiently.