Question

How do you add these two consecutive triangular numbers? TR(4)+TR(3)

Answer 1

The n-th trianglar number is given by

$T_n = \dfrac{n(n+1)}2$

Then

$T_4 + T_3 = \dfrac{4\times5}2 + \dfrac{3\times4}2 = \boxed{16}$

Alternatively, the triangular numbers are defined recursively by

$\begin{cases}T_1 = 1 \\ T_n = T_{n-1} + n & \text{for } n>1\end{cases}$

This means

$T_1=1$

$T_2 = T_1 + 2 = 3$

$T_3 = T_2 + 3 = 6$

$T_4 = T_3 + 4 = 10$

so that

$T_3+T_4=\boxed{16}$