Question

What square numbers will we get if we add: a) 8th and 9th triangular numbers b) 13th and 14th triangular numbers

Answer 1

Dear student,

The general formula of the triangular number is,
$$\begin{gathered} \frac{n\times (n+1)}{2} \\ Where,n=1,2,3,4,..... \\ a)Sothe8thtriangluarnumberis, \\ \frac{{\displaystyle 8\times (8+1)}}{{\displaystyle 2}}=36 \\ 9thtriangularnumberis, \\ \frac{{\displaystyle 9\times (9+1)}}{{\displaystyle 2}}=45 \\ Theirsumis, \\ 36+45=81 \\ Whichisaperfectsquareof9. \\ Sotheansweris81. \\ b)Sothe13thtriangluarnumberis, \\ \frac{{\displaystyle 13\times (13+1)}}{{\displaystyle 2}}=91 \\ 9thtriangularnumberis, \\ \frac{{\displaystyle 14\times (14+1)}}{{\displaystyle 2}}=105 \\ Theirsumis, \\ 91+105=196 \\ Whichisaperfectsquareof14. \\ Sotheansweris196. \end{gathered}$$
Regards