Discussion

Find the value of \(5^3 + 6^3 + 7^3 + ... + 10^3\).

Answer: A

We use the formula for the sum of the cubes of the first n natural numbers: \(S_n = [\frac{n(n+1)}{2}]^2\).

The required sum is (Sum of cubes from 1 to 10) - (Sum of cubes from 1 to 4).

Sum of cubes from 1 to 10 = \([\frac{10(10+1)}{2}]^2 = (\frac{110}{2})^2 = 55^2 = 3025\).

Sum of cubes from 1 to 4 = \([\frac{4(4+1)}{2}]^2 = (\frac{20}{2})^2 = 10^2 = 100\).

The required sum = \(3025 - 100 = 2925\).

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