Question

Express $25$ as the sum of triangular numbers.

Answer 1

Hint: Triangular numbers are figurate numbers which means that these are the numbers of the objects that can be arranged in an equilateral triangle shape. The examples of the triangular numbers are-
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, …
The formula used to calculate the triangular number is given by-
${T_n} = \dfrac{{n\left( {n + 1} \right)}}{2}$
Where, ${T_n}$ is the triangular number and $n = 0,1,2,3,...$

Complete step-by-step answer:
The sum of the triangular numbers ${S_n} = 25$
Now we have to calculate the triangular numbers using the formula starting from $n = 0$, we have,
At $n = 0$
$\begin{array}{c}
{T_0} = \dfrac{{0\left( {0 + 1} \right)}}{2}\\
 = 0
\end{array}$
So, the first triangular number is $0$.
At $n = 1$
$\begin{array}{c}
{T_1} = \dfrac{{1\left( {1 + 1} \right)}}{2}\\
 = \dfrac{2}{2}\\
 = 1
\end{array}$
So, the second triangular number is $1$.
At $n = 2$
$\begin{array}{c}
{T_2} = \dfrac{{2\left( {2 + 1} \right)}}{2}\\
 = \dfrac{6}{2}\\
 = 3
\end{array}$
So, the third triangular number is $3$.
At $n = 3$
$\begin{array}{c}
{T_3} = \dfrac{{3\left( {3 + 1} \right)}}{2}\\
 = \dfrac{{12}}{2}\\
 = 6
\end{array}$
So, the fourth triangular number is $6$.
At $n = 4$
$\begin{array}{c}
{T_4} = \dfrac{{4\left( {4 + 1} \right)}}{2}\\
 = \dfrac{{20}}{2}\\
 = 10
\end{array}$
So, the fifth triangular number is $10$.
At $n = 5$
$\begin{array}{c}
{T_5} = \dfrac{{5\left( {5 + 1} \right)}}{2}\\
 = \dfrac{{30}}{2}\\
 = 15
\end{array}$
So, the sixth triangular number is $15$.
Now we have obtained the triangular numbers whose sum is equals to $25$,the triangular numbers whose sum is $25$ are as follows-
${S_n} = {T_0} + {T_4} + {T_5}$
Or we can write this as-
$25 = 0 + 10 + 15$
Or,
$25 = 10 + 15$
Therefore, $25$ is the sum of two triangular numbers $10{\rm{ and 15}}$.

Note: It should be noted that we need to calculate only those triangular numbers whose sum when added would be equal to $25$ that is why, we have calculated a total of $5$ triangular numbers. In those $5$ triangular numbers we have obtained $10{\rm{ and 15}}$, that was our answer.
The alternative method was to directly use the values of triangular numbers $10{\rm{ and 15}}$ and find the sum. For example, $10$ and $15$ both are the triangular numbers and their sum is given by,
$25 = 10 + 15$