Question 141
Consider the table of numbers below. The first line has the numbers from 1 to $n$. The second has the numbers from 1 to $n$ with each multiplied by 2. The lines follow this pattern to the last line that presents $n$ times each number from 1 to $n$. 
We will use it to calculate the value of the expression $$
1^{3}+2^{3}+\ldots+100^{3}
$$ In addition to the table, we will use the fact that $$
1+2+\ldots+n=\frac{n(n+1)}{2}
$$
(a) Determine the sum of all the numbers of the number line $k$. With this, determine an expression for the sum of all numbers in the table.
(b) Note pieces in the table separating it into layers in the form of $L$. The numbers in a certain layer $k$ are: $k, 2 k, \ldots,(k-1) k, k^{2},(k-1) k, \ldots, 2 k, k$. Determine the sum of the numbers of this layer as a function of $k$.
(c) Adding the results of all layers, we will arrive at the same result as by adding all the lines. Adding this information Determine the expression value: $$
1^{3}+2^{3}+\ldots+100^{3}
$$
We will use it to calculate the value of the expression $$1^{3}+2^{3}+\ldots+100^{3}
$$ In addition to the table, we will use the fact that $$
1+2+\ldots+n=\frac{n(n+1)}{2}
$$
(a) Determine the sum of all the numbers of the number line $k$. With this, determine an expression for the sum of all numbers in the table.
(b) Note pieces in the table separating it into layers in the form of $L$. The numbers in a certain layer $k$ are: $k, 2 k, \ldots,(k-1) k, k^{2},(k-1) k, \ldots, 2 k, k$. Determine the sum of the numbers of this layer as a function of $k$.
(c) Adding the results of all layers, we will arrive at the same result as by adding all the lines. Adding this information Determine the expression value: $$
1^{3}+2^{3}+\ldots+100^{3}
$$