SOLUTION: what is the one millionth term in the sequence 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5.....in which each positive integer n occurs n times please help me to solve this!!! i really don't

Algebra ->  Sequences-and-series -> SOLUTION: what is the one millionth term in the sequence 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5.....in which each positive integer n occurs n times please help me to solve this!!! i really don't      Log On


   

Question 194091: what is the one millionth term in the sequence 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5.....in which each positive integer n occurs n times
please help me to solve this!!!
i really don't know how to solve
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First, let's number each term and denote these numbers as "n". In other words, 1 is the first term, 2 is the second term, 2 is the third term, 3 is the fourth term, etc... to get 

n:    1,  2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12,  13,  14,  15
term: 1,  2,  2,  3,  3,  3,  4,  4,  4,   4,   5,   5,   5,   5,   5



Notice how the LAST number of each grouping lands on the "n" values: 1, 3, 6, 10, and 15

This sequence is the list of triangular numbers which can be found through the formula %28m%28m%2B1%29%29%2F2

So for example, the last occurrence of the number 3 is when n=6. It turns out that %283%283%2B1%29%29%2F2=%283%2A4%29%2F2=12%2F2=6.

Also, the last occurrence of the number 4 is when n=10. It turns out that %284%284%2B1%29%29%2F2=%284%2A5%29%2F2=20%2F2=10.

etc...


Now use the same formula to find where the LAST occurrence of 6 will be:

%286%286%2B1%29%29%2F2=%286%2A7%29%2F2=42%2F2=21

So this means that the LAST occurrence of the number 6 will be when n=21 (ie it will be the 21st term). Note: write out the next 6 terms and you'll find that the LAST 6 does fall on n=21


You can continue this indefinitely. So this means that we can use this formula to find out what the millionth term of the sequence is.


It turns out that if you plug in m=1414, you get %281414%281414%2B1%29%29%2F2=%281414%281415%29%29%2F2=2000810%2F2=1000405. This means that the LAST occurrence of the term 1414 is in the location n=1000405. Since 405 is smaller than 1,000,000, this means that the millionth term is also the value 1414.

Note: to find the value 1414, you can either guess or solve the equation %28m%28m%2B1%29%29%2F2=1000000 to find an approximate value of "m". From there, round to the nearest whole number.


Answer:


So the millionth term is 1414.