Consider the table of numbers below. The first row contains the numbers from 1 to $n$. The second row contains the numbers from 1 to $n$, each multiplied by 2. The rows follow this pattern until the last row, which presents each number from 1 to $n$ multiplied by $n$.

![](https://cdn.mathpix.com/cropped/2024_05_01_40ea5c1759d4b041c087g-13.jpg?height=597&width=537&top_left_y=678&top_left_x=859)

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 in row $k$. With this, determine an expression for the sum of all the numbers in the table.

(b) Observe pieces in the table by separating it into layers in the shape of an $L$. The numbers in a certain layer $k$ are: $k, 2k, \ldots, (k-1)k, k^2, (k-1)k, \ldots, 2k, k$. Determine the sum of the numbers in this layer in terms of $k$.

(c) By summing the results of all the layers, we will arrive at the same result as summing all the rows. Combining this information, determine the value of the expression:

$$
1^{3}+2^{3}+\ldots+100^{3}
$$
Solution

(a) The numbers in row $t$ are the numbers from 1 to $n$ multiplied by $t$. The sum of them is

$$
\begin{aligned}
t+2 t+3 t+\ldots+n t & =t(1+2+3+\ldots+n) \\
& =t \frac{n(n+1)}{2}
\end{aligned}
$$

The sum of the numbers in the table can be calculated using the sum of all rows. Since $\frac{n(n+1)}{2}$ is a common factor, it can be factored out, and we will have

$$
\frac{n(n+1)}{2}(1+2+3+\ldots+n)=\left(\frac{n(n+1)}{2}\right)^{2}
$$

(b) Adding the numbers we get:

$$
\begin{aligned}
k+2 k+\ldots+(k-1) k+k^{2}+(k-1) k+\ldots+k & =k^{2}+2 k(1+2+3+\ldots+(k-1)) \\
& =k^{2}+2 k \frac{(k-1) k}{2} \\
& =k^{2}+k^{2}(k-1) \\
& =k^{3}
\end{aligned}
$$

(c) If we create the table for $n=100$, we will have 100 rows and 100 layers. With the information from the previous items, we can conclude that

$$
\begin{aligned}
1^{3}+2^{3}+\ldots+100^{3} & =\left(\frac{100 \cdot 101}{2}\right)^{2} \\
& =5050^{2} \\
& =25502500
\end{aligned}
$$