Find the total volume of 15 cubes whose edges are 16 cm, 17 cm, 18 cm, …., 30 cm respectively

Given that there are 15 cubes,

We see that their sides 16cm, 17cm..,30cm are in series.

To find the total volume, we know that the volume of a cube is simply l³ where l is the length of the side.

Volume of first cube = 16cm × 16cm × 16cm = 16³ cm³

Volume of second cube = 17cm × 17cm × 17cm = 17³ cm³

And so on.

We observe that this is in a series.

So S = 16³ + 17³ + 18³ + … + 30³

To find S,

Let S₁ be 1³ + 2³ + …. + 30³ with n = 30

Let S₂ be 1³ + 2³ + …. + 15³ with n = 15

S = S₁-S₂

To find S₁

1³ + 2³ + …. + 30³ with n = 30

Formula to find the sum of first n cubes of natural numbers is

S =

S₁ =

=

S₁ = 216225

To find the sum S₂ –

1³ + 2³ + …. + 15³ with n = 15

Formula to find the sum of first n cubes of natural numbers is

S =

S₂ =

=

S₂ = 14400

S = S₁-S₂

S = 216225-14400

S = 201825 cm³

The required sum S = 16³ + 17³ + 18³ + … + 30³ = 201825 cm³