Question

Find the $14$-th and $15$-th triangular numbers, and find their sum.

Answer 1

The $n^{th}$ triangular number is equal to the

sum of $n$ natural number from $1$ to $n$.

this is denoted by $T_n=\sum _{x=1}^{n}k=1+2+3+....+n$

$=\frac{n(n+1)}{2}$

${14}^{th}$ triangular number is $T_{14}=\frac{14(14+1)}{2}=7\times 15=105$

${15}^{th}$ triangular number is $T_{15}=\frac{15(15+1)}{2}=15\times 8=120$

Sum of the ${14}^{th}$ and ${15}^{th}$ triangular number is

$=T_{14}+T_{15}=105+120=225$ Ans.