For each positive integer n, the nth positive triangular num

[fskilnik@GMATH]

GMATH practice exercise (Quant Class 17)



For each positive integer n, the nth positive triangular number is equal to the number of dots in the triangular arrangement with n dots on a side. The first positive triangular numbers are 1, 3, 6, 10, 15, and 21, as shown. Which of the following numbers is NOT a positive triangular number?

(A) 105
(B) 210
(C) 300
(D) 311
(E) 378

Answer: [spoiler]___(D)__[/spoiler]

[fskilnik@GMATH]

GMATH practice exercise (Quant Class 17)



For each positive integer n, the nth positive triangular number is equal to the number of dots in the triangular arrangement with n dots on a side. The first positive triangular numbers are 1, 3, 6, 10, 15, and 21, as shown. Which of the following numbers is NOT a positive triangular number?

(A) 105
(B) 210
(C) 300
(D) 311
(E) 378

$${T_n} = 1 + 2 + \ldots + n\,\,\mathop = \limits^{\left( * \right)} \,\,{{n \cdot \left( {n + 1} \right)} \over 2}\,\,\,\,\,\,\,\left( {n \ge 1\,\,{\mathop{\rm int}} } \right)\,\,\,\,\,\,\,\,\,\,\,\left[ {\left( * \right)\,\,{\rm{arithmetic}}\,\,{\rm{sequence}}} \right]$$
$$?\,\,\,:\,\,\,\underline {{\rm{not}}} \,\,\,{T_n}$$

$$\left( A \right)\,\,\left\{ \matrix{
\,n \cdot \left( {n + 1} \right) = 2 \cdot 105 = 210 \hfill \cr
\,15 \cdot 15 = 225 \hfill \cr} \right.\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{try}}!} \,\,\,\,{{14 \cdot 15} \over 2} = 7 \cdot 15 = 105\,\,\,{\rm{works}}!\,\,\,\,\, \Rightarrow \,\,\,\,\,105 = {T_{14}}$$
$$\left( B \right)\,\,\left\{ \matrix{
\,n \cdot \left( {n + 1} \right) = 2 \cdot 210 = 420 \hfill \cr
\,20 \cdot 20 = 400 \hfill \cr} \right.\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{try}}!} \,\,\,\,{{20 \cdot 21} \over 2} = 10 \cdot 21 = 210\,\,\,{\rm{works}}!\,\,\,\,\, \Rightarrow \,\,\,\,\,210 = {T_{20}}$$
$$\left( C \right)\,\,\left\{ \matrix{
n \cdot \left( {n + 1} \right) = 2 \cdot 300 = 600 \hfill \cr
25 \cdot 25 = 625 \hfill \cr} \right.\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{try}}!} \,\,\,\,{{24 \cdot 25} \over 2} = 12 \cdot 25 = 300\,\,\,{\rm{works}}!\,\,\,\,\, \Rightarrow \,\,\,\,\,300 = {T_{24}}$$
$$\left( D \right)\,\,\,{T_{24}} < 311 < 300 + 25 = {T_{24}} + 25 = {T_{25}}$$


The correct answer is (D).


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.