What is the 2005th term of the sequence ?

 

 

We need to find a way to count the terms.  There are one 1’s, two 2’s, three 3’s and so on.  After the 1’s, there is 1 number, after the 2’s there are 3 numbers (1 + 2), after the 3’s, there are 6 numbers (1 + 2 + 3), and so forth.  The number of terms (numbers in the list) after n numbers have been used is the triangular number .

 

We can then find the 2005^th term by solving the equation .  Since this equation has no rational roots, we can approximate the answer and then find the correct answer by guessing and checking.

 

 

A good estimate for the solution to this quadratic equation would be .

 

If we try 63 in our formula it will give us the number of terms after 63 is used.  .  Since 63 is used 63 times, all the terms from 2016 – 63 = 1953 to 2016 are 63.  2005 is in this interval, so the answer is 63.