SOLUTION: A sequence goes 1,2,2,3,3,3,... Find the 66th term. Please show all the steps and explain why it is like that

Question 371140: A sequence goes 1,2,2,3,3,3,...
Find the 66th term. Please show all the steps and explain why it is like that

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
The number 1 is the first term.

The last number of 2 is term number 2+1=3

The last number of 3 is term number 3+2+1=6

The last number of 4 is term number 4+3+2+1=10 (expand it out to get 1,2,2,3,3,3,4,4,4,4 and you'll see that the last 4 is the 10th term)

The last number of 5 is term number 5+4+3+2+1=15 (expand it out to get 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5 and you'll see that the last 5 is the 15th term)

This pattern continues....

If you keep going, you'll get the sequence: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55

(ie the 21 refers to the position of the last 6)

Now because 1+2+3+4+5+6+7+8+9+10+11 = 66, this means that the last 11 will be in the 66th position.

So the 66th term is 11.

To verify, list out the terms to get

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

and if you count each term, you'll find that the last term in that list is indeed the 66th term (which is 11)

Hopefully, this makes sense. If not, please let me know.
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