Triangular Numbers 4 – Find A Formula

We know that the rectangle formed from two of the $n$ th triangular numbers has dimensions $n\times n+1$ . Thus, the size of just one of the triangular numbers is $\dfrac{n\times (n+1)}{2}$ , or $\boxed{\dfrac{n(n+1)}{2}}$ .

1,3,6,10..... now if we put the values of n in the equation n(n+1)/2 , the same sequence formed.

Observing the rows and columns of the rectangles , the series goes likes this 1-(1.2), 2-(2.3),3-(3.4),4(4.5),.........................n-{n(n+1)}.The half of the rectangle gives triangle(Those dark circles) so n(n+1)/2

The triangles in the pattern are formed by adding successive numbers.

So, the formula to find the number of dots in the $n^{th}$ triangular number is:

$= \frac{n(n+1)}{2}$ (Formula to find sum of numbers from $1$ to $n$ )

So, the answer is: $\boxed{\frac{n(n+1)}{2}}$

The formula to derive the nth rectangular number is n(n+1). Now divide it by two and we get the general formula of the nth triangular number.

As simple as the area of rectangle divided by 2.

the series is like 1,3,6,10,15..........f(5)=5(5+1)/2=15