<p>1,2,2,3,3,3,4,4,4,4,......</p>
<p>All Positive integers appear in the sequence above,
and each positive integer k appears in the sequence
k times. In the sequence, each term after the first is
greater than or equal to each of the terms before it.
If the integer 12 first appears in the sequence as the
nth term, what is the value of n ?</p>
<p>I would think of triangle numbers. As one can see fairly quickly, the first occurrence of x appears at triangle(x-1) +1. The formula for the kth triangular number is k(k+1)/2.
For x = 12, you get 11*12/2 +1 = 67.</p>
<p>well you know that 12 appears after 1+2+3+4…till 11 and then you add 1 for the 12 since you know each number k, appears k times.</p>
<p>I’d personally keep it simple and do:
[quote]
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11<a href=“honestly%20took%20me%20less%20than%2010%20seconds%20to%20plug%20in%20on%20my%20calculator”>/quote</a></p>
<p>Which yields 66.
Since 12 is the first 12 to appear = 67th term.</p>