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# For each positive integer n, the nth positive triangular number is equ

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GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 910
For each positive integer n, the nth positive triangular number is equ  [#permalink]

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24 Feb 2019, 13:36
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Difficulty:

65% (hard)

Question Stats:

45% (02:35) correct 55% (02:02) wrong based on 20 sessions

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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

_________________

Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net

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Re: For each positive integer n, the nth positive triangular number is equ  [#permalink]

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24 Feb 2019, 15:01
So study these numbers carefully. You will fibd that T(n)= Sum of all natural numbers from 1 to n. We know that this sum is given by the general formula (n(n+1))/2.

So one method can be assuming that each of these options is given by such a formula and trying to find a value of n that satisfies it. The one for which this is not possible is your answer. Right now i can’t think of any other method off the cuff. So let me fasten up the above method for you. I’ll update here if i think of some other method.

Assume each option = n(n+1)/2
So 2* each option = product of 2 consecutive nunbers. Let’s consider each option now.

As you browse the options you’ll find that 311 is prime so 311*2 can never be expressed as the sume of 2 consecutive numbers so the answer is D

Cheers

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Re: For each positive integer n, the nth positive triangular number is equ  [#permalink]

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25 Feb 2019, 08:27
fskilnik wrote:
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

values of the triangular no are given n*(n+1)/2
so out of given options 311 is prime no. which is not possible
IMO D
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Re: For each positive integer n, the nth positive triangular number is equ  [#permalink]

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25 Feb 2019, 12:39
fskilnik wrote:
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

$${T_n} = 1 + 2 + \ldots + n\,\,\,\,\left( {n \ge 1\,\,{\mathop{\rm int}} } \right)$$

$${T_n} = {{n \cdot \left( {n + 1} \right)} \over 2}\,\,\,\,\,\,\left[ {{\rm{arithmetic}}\,\,{\rm{sequence}}} \right]$$

$$?\,\,\,:\,\,\,\underline {{\rm{not}}} \,\,\,T$$

$$\left( A \right)\,\,\left\{ \matrix{ \,n \cdot \left( {n + 1} \right) = 2 \cdot 105 = 210 \hfill \cr \,15 \cdot 15 = 225 \hfill \cr} \right.\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{try}}!} \,\,\,\,{{14 \cdot 15} \over 2} = 7 \cdot 15 = 105\,\,\,{\rm{works}}!\,\,\,\,\, \Rightarrow \,\,\,\,\,105 = {T_{14}}$$

$$\left( B \right)\,\,\left\{ \matrix{ \,n \cdot \left( {n + 1} \right) = 2 \cdot 210 = 420 \hfill \cr \,20 \cdot 20 = 400 \hfill \cr} \right.\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{try}}!} \,\,\,\,{{20 \cdot 21} \over 2} = 10 \cdot 21 = 210\,\,\,{\rm{works}}!\,\,\,\,\, \Rightarrow \,\,\,\,\,210 = {T_{20}}$$

$$\left( C \right)\,\,\left\{ \matrix{ n \cdot \left( {n + 1} \right) = 2 \cdot 300 = 600 \hfill \cr 25 \cdot 25 = 625 \hfill \cr} \right.\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{try}}!} \,\,\,\,{{24 \cdot 25} \over 2} = 12 \cdot 25 = 300\,\,\,{\rm{works}}!\,\,\,\,\, \Rightarrow \,\,\,\,\,300 = {T_{24}}$$

$$\left( D \right)\,\,\,{T_{24}} < 311 < 300 + 25 = {T_{24}} + 25 = {T_{25}}$$

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

Regards,
Fabio.
_________________

Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net

Re: For each positive integer n, the nth positive triangular number is equ   [#permalink] 25 Feb 2019, 12:39
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