The value of \({1}^{3}-{2}^{3}+{3}^{3}-\ldots +{15}^{3}\) is: [JEE Main 2026, 6 Apr (Shift 1)]

\(S=1^3-2^3+3^3-\ldots+15^3\)

\(=\left(1^3+2^3+\ldots+15^3\right)-2\left(2^3+4^3+\ldots+14^3\right)\)

\(=\left(1^3+2^3+\ldots+15^3\right)-2^4\left(1^3+2^3+\ldots+7^3\right)\)

\(=\left[\frac{15 \times 16}{2}\right]^2-16\left[\frac{7 \times 8}{2}\right]^2\)

\(=14400-16\times 784\)

\(=14400-12544\)

\(=1856\)