For positive integer m, the m-th heptagonal number is given

Question

For positive integer m, the m-th heptagonal number is given by the formula (5m2 – 3m)/2. For positive integer n, the n-th triangular number is the sum of the first n positive integers. Which of the following is true for k, the smallest triangular number that is also heptagonal?
(A) 33 ≤ k ≤ 40
(B) 41 ≤ k ≤ 48
(C) 49 ≤ k ≤ 56
(D) 57 ≤ k ≤ 64
(E) 65 ≤ k ≤ 72

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(A) 33 ≤ k ≤ 40
(B) 41 ≤ k ≤ 48
(C) 49 ≤ k ≤ 56
(D) 57 ≤ k ≤ 64
(E) 65 ≤ k ≤ 72

Bunuel, can you please help with this one?

This question was moved to an archive because this concept is NOT tested on the GMAT
Original tags were MGMAT, Algebra, Too hard, and 700+

Bunuel · Answer

It's been a long time since I've last heard about heptagonal and triangular numbers. Anyway, probably the best way would be to write down the numbers.

Since the n-th triangular number is the sum of the first n positive integers, then n-th triangular number is given by the formulas n(n+1)/2.

For example:
The 1st triangular number is 1(1+1)/2=1;
The  2 nd triangular number is 2(2+1)/2=3=1+ 2 ;
The  3 rd triangular number is 3(3+1)/2=6=3+ 3 ;
The  4 th triangular number is 6+ 4 =10;
The  5 th triangular number is 10+ 5 =15;
...

So, triangular numbers are:  1 , 1+ 2 =3, 3+ 3 =6, 6+ 4 =10, 10+ 5 =15, 15+ 6 =21, 21+ 7 =28, 28+ 8 =36, 36+ 9 =45, 45+ 10 = 55 , 55+ 11 =66, ...

On the other hand, heptagonal numbers are:  1 , 7, 18, 34,  55 , 81, ... using (5m^2 – 3m)/2.

So, as you can see the smallest triangular number that is also heptagonal is 1. Since it's not among answer choices I guess they don't consider 1, so the next one is 55.

Answer: C.

rajareena · Answer

Hi Bunuel

Do you think this particular concept is tested on GMAT?

Bunuel · Answer

No, this concept is not tested on the GMAT.

coolpintu · Answer

The triangular number = n(n+1)/2  .

The 1st triangular number is 1(1+1)/2=1;
The 2nd triangular number is 2(2+1)/2=3=1+2;
The 3rd triangular number is 3(3+1)/2=6=3+3;
The 4th triangular number is 6+4=10;
The 5th triangular number is 10+5=15;

So, triangular numbers are: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66......

On the other hand, heptagonal numbers are: 1, 7, 18, 34, 55, 81, ... using (5m^2 – 3m)/2.

So, the smallest triangular number that is also heptagonal is 1 which is not present in any choice. Hence, look for the next one. So the next one is 55.
Only C satisfied the equation.

Answer: C.

BangOn · Answer

If m = 1, then the heptagonal number is (5*1^2 – 3×1)/2 = (5 – 3)/2 = 1.
If m = 2, then the heptagonal number is (5*2^2 – 3×2)/2 = (20 – 6)/2 = 14/2 = 7.
If m = 3, then the heptagonal number is (5*3^2 – 3×3)/2 = (45 – 9)/2 = 36/2 = 18.
If m = 4, then the heptagonal number is (5*4^2 – 3×4)/2 = (80 – 12)/2 = 68/2 = 34.
If m = 5, then the heptagonal number is (5*5^2 – 3×5)/2 = (125 – 15)/2 = 110/2 = 55.
If m = 6, then the heptagonal number is (5*6^2 – 3×6)/2 = (180 – 18)/2 = 162/2 = 81.

Using n*(n+1)/2
if n =1, triangle number is 1
if n =2, triangle number is 3
if n =3 triangle number is 6
if n =4, triangle number is 10
if n =5, triangle number is 15
if n =6 triangle number is 21
if n=7, triangle number is 28
if n = 8, triangle number is 36
if n =9, triangle number is 45
if n = 10 triangle is 55.....stop...

We have 55 as the answer.

the target number must be 34 or 55, A or C now.
Now we need to find the smallest value which will exist for both triangle and heptagonal

krrish · Answer

hey thanks for the beautiful question

i dont know about questions of this kind till date.. thanks again a here's my approach : heptagonal numbers : 5m^2 -3m from choices we can see the total range is only till 72 so first wirte down these by substituting values of k = 1,2, 3 we get 1,7, 18, 34, 5x 22/2 = 55, 27x3 we have reached till 72 out limit so we can't exceed from here now get to 2nd one (nx n+1)/2 so u can easly get 55 by keeping n = 10

and our answer is C

hope this helps, please PM for any queries

samusa · Answer

Another approach....though I think that solution by calculating individual series is a faster method. the m-th heptagonal number = n-th triangular number= k (5m^2 – 3m)/2 = n ( n+1) /2 = k (5m^2 – 3m) = n ( n+1) = 2 k n ( n+1) = 2K = Product of two consecutive integers. 33 ≤ k ≤ 40 So 66 ≤2 k ≤ 80 so 8 x 9 = 72 41 ≤ k ≤ 48 So 82 ≤2 k ≤ 96 so 9 x10 = 90 49 ≤ k ≤ 56 So 98 ≤2 k ≤ 112 so 10 x 11 =110 57 ≤ k ≤ 64 So 114≤ 2k ≤ 128 nothing lies in between... . 65 ≤ k ≤ 72 So 130≤ 2k ≤ 144 so 11 x 12 =132 so now we have to check ( cumbersome ... but easy.) 5m^2 – 3m- 2 k = 0 The only value of 2k that satisfies the above equation is 110. So K= 55 Hence, C. :-D

rhine29388 · Answer

What do you mean exactly by the terms ' heptagonal number ' and ' triangular number ' ?

bumpbot · Answer

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For positive integer m, the m-th heptagonal number is given

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