Enter the starting and ending numbers to quickly calculate the sum of squares and cubes of consecutive numbers.
The sum of squares and cubes refers to the total of the square or cube values of consecutive numbers. Based on the starting and ending numbers, you can either calculate each number's square or cube individually and then add them up, or use a formula to handle it more efficiently.
If the starting number \(a\) and ending number \(b\) are known, the sum of squares can be calculated by squaring each number in the range and then adding them up. The formula is as follows:
\( S_{\text{square}} = a^2 + (a+1)^2 + (a+2)^2 + ... + b^2 \)
The calculation for the sum of cubes follows a similar process. You can calculate each cube and then sum them using the formula:
\( S_{\text{cube}} = a^3 + (a+1)^3 + (a+2)^3 + ... + b^3 \)
The integers from 5 to 10 are 5, 6, 7, 8, 9, and 10. Their sum of squares is: \( 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 \) Calculating each square: \( 25 + 36 + 49 + 64 + 81 + 100 = 355 \) Conclusion: The sum of squares for integers from 5 to 10 is 355.
The odd numbers from 3 to 7 are 3, 5, and 7. Their sum of squares is: \( 3^2 + 5^2 + 7^2 \) Calculating each square: \( 9 + 25 + 49 = 83 \) Conclusion: The sum of squares for odd numbers from 3 to 7 is 83.
The even numbers from 10 to 14 are 10, 12, and 14. Their sum of cubes is: \( 10^3 + 12^3 + 14^3 \) Calculating each cube: \( 1000 + 1728 + 2744 = 5472 \) Conclusion: The sum of cubes for even numbers from 10 to 14 is 5472.