Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:
This problem isn’t conceptually too difficult – it just takes quick, accurate computation. First, work out the first several heptagonal numbers.
If m = 1, then the heptagonal number is (5×12 – 3×1)/2 = (5 – 3)/2 = 1.
If m = 2, then the heptagonal number is (5×22 – 3×2)/2 = (20 – 6)/2 = 14/2 = 7.
If m = 3, then the heptagonal number is (5×32 – 3×3)/2 = (45 – 9)/2 = 36/2 = 18.
If m = 4, then the heptagonal number is (5×42 – 3×4)/2 = (80 – 12)/2 = 68/2 = 34.
If m = 5, then the heptagonal number is (5×52 – 3×5)/2 = (125 – 15)/2 = 110/2 = 55.
If m = 6, then the heptagonal number is (5×62 – 3×6)/2 = (180 – 18)/2 = 162/2 = 81.
We can stop here, because we’re above the range in the answer choices. In fact, the target number must be 34 or 55, so we could narrow down to A or C now.
Next, which of these two numbers is triangular: the sum of the first n positive numbers? Work these triangular numbers out:
1
1 + 2 = 3
1 + 2 + 3 = 6
plus 4 gives 10
plus 5 gives 15
plus 6 gives 21
plus 7 gives 28
plus 8 gives 36 – we can stop here, because we’ve skipped over 34. The answer must be 55 (as you can show by adding 9 and 10).
The correct answer is C.