Question

Find the ${14^{th}}$ and ${15^{th}}$ triangular numbers, and find their sum.

Answer 1

Hint: In this question, we’ve asked the${14^{th}}$ and ${15^{th}}$ triangular numbers, As the nth triangular number is given by \[{\text{ }}1 + 2 + 3 + 4 + 5 + 6 + 7 + \ldots . + n\] $ = \dfrac{{n\left( {n + 1} \right)}}{2}$, so using this formula of the nth triangular number we’ll find the value of ${14^{th}}$ and ${15^{th}}$ triangular numbers.
After getting their values we’ll simply add their values to get the answer.

Complete step-by-step answer:
We know that the nth triangular number \[ = {\text{ }}1 + 2 + 3 + 4 + 5 + 6 + 7 + \ldots . + n\]
Also, it is well known that,
$1 + 2 + 3 + ....n = \dfrac{{n\left( {n + 1} \right)}}{2}$
Therefore we can say that nth triangular number $ = \dfrac{{n\left( {n + 1} \right)}}{2}$
On Substituting n=14, we get,
${14^{th}}$triangular number$ = \dfrac{{14\left( {14 + 1} \right)}}{2}$
On Simplifying the terms in brackets, we get,
$ = \dfrac{{14\left( {15} \right)}}{2}$
On Dividing numerator and denominator by 2, we get,
$ = 7\left( {15} \right)$
On Simplifying the brackets, we get,
$ = 105$
Now, Substituting n=15, in
nth triangular number $ = \dfrac{{n\left( {n + 1} \right)}}{2}$, we get,
${15^{th}}$triangular number$ = \dfrac{{15\left( {15 + 1} \right)}}{2}$
On Simplifying the terms in brackets, we get,
$ = \dfrac{{15\left( {16} \right)}}{2}$
On Dividing numerator and denominator by 2, we get,
$ = 15\left( 8 \right)$
On Simplifying the brackets, we get,
$ = 120$
Therefore the ${14^{th}}$ and ${15^{th}}$ triangular numbers are 105 and 120 respectively
Now the sum of ${14^{th}}$ and ${15^{th}}$ triangular numbers$ = 105 + 120$
On Adding both the values,
The sum of ${14^{th}}$ and ${15^{th}}$ triangular numbers$ = 225$.

Note: As calculation is involved in this question one should avoid making calculation mistakes. Also instead of adding 1 to 14 or 1 to 15 we use the formula, $1 + 2 + 3 + ....n = \dfrac{{n\left( {n + 1} \right)}}{2}$. This makes the answer simpler and we can avoid calculation mistakes.
A triangular number or triangle number is the number of dots required to make an equilateral triangle. A nth triangular number will be the number of dots required to complete a triangle with n dots at every side as well as having n number of rows arranged in such an order that nth row will contain only n number of dots.
For example the fifth triangular number series will be