Question

4^3+5^3+6^3+....+15^3​

Answer 1

We use the formula,

$\longrightarrow\boxed{1^3+2^3+3^3+\,\dots\,+n^3=\left(\dfrac{n(n+1)}{2}\right)^2}$

Here,

$\longrightarrow S=4^3+5^3+6^3+\,\dots\,+15^3$

$\longrightarrow S=(1^3+2^3+3^3+\,\dots\,+15^3)-(1^3+2^3+3^3)$

$\longrightarrow S=\left(\dfrac{15\times16}{2}\right)^2-\left(\dfrac{3\times4}{2}\right)^2$

$\longrightarrow S=120^2-6^2$

$\longrightarrow S=14400-36$

$\longrightarrow\underline{\underline{S=14364}}$

Hence 14364 is the answer.

Answer 2

Sum of cubes of first $\sf n$ natural numbers is given by,

$\green{\bigstar} \ \ \boxed{\sf{1^3 + 2^3 + 3^3 + 4^3 + \dots + n^3 = \Bigg[\dfrac{n(n+1)}{2}\Bigg]^2}}$

$\sf{\\}$

Let, $\sf S = 4^3 + 5^3 + 6^3 + \dots + 15^3$

$\longrightarrow \scriptsize \sf S = (1^3 + 2^3 + 3^3 + 4^3 +\dots+15^3) - (1^3 + 2^3 + 3^3)$

$\longrightarrow \sf S = \Bigg[\dfrac{15(15+1)}{2}\Bigg]^2 - \Bigg[\dfrac{3(3+1)}{2}\Bigg]^2$

$\longrightarrow \sf S = (15 * 8)^2 - (3 * 2)^2$

$\longrightarrow \sf S = 120^2 - 6^2$

$\longrightarrow \sf S = 14400 - 36$

$\longrightarrow \sf S = 14364$

$\sf{\\}$

Hence, the answer is,

$\longrightarrow \underline{\underline{\sf{\green{4^3 + 5^3+6^3+\dots+15^3 = 14364}}}}$